Finite semisimple group algebra of a normally monomial group
Shalini Gupta Sugandha Maheshwary
math.RA 16S34, 20C05, 16K20

	18pt
	
		
	
	
	Finite semisimple group algebra of a normally monomial group
    Shalini Gupta [Corresponding author] 
 Department of Mathematics, 

		Punjabi University, Patiala,  India.
email: shalini@pbi.ac.in Sugandha Maheshwary [Research supported by SERB, India (PDF/2016/000731).]
		
 Department of Mathematical Sciences,
Indian Institute of Science Education and Research, Mohali,

		Sector 81, Mohali (Punjab)-140306, India.
		
email: sugandha@iisermohali.ac.in
	
    
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	In this paper, the complete algebraic structure of finite semisimple group algebra of a normally monomial group is described. The main result is illustrated by computing the explicit Wedderburn decomposition of finite semisimple group algebras of various normally monomial groups. The automorphism groups of these group algebras are also determined.


	Keywords : semisimple group algebra, normally monomial groups, primitive central idempotents,  Wedderburn decomposition. 
 

	MSC2000 : 16S34; 20C05; 16K20
	
	
	

§  INTRODUCTION
Let 𝔽_q denote the field containing q elements and let G be a finite group of order relatively prime to q, so that 
the group algebra 𝔽_qG is semisimple. The knowledge of the algebraic structure of 𝔽_qG has applications in coding theory and is useful in describing the automorphism group as well as the unit group of 
𝔽_qG. This has attracted the attention of several authors <cit.>. Broche et al. <cit.> gave  description of semisimple group algbra 𝔽_qG, when G is an abelian-by-supersolvable group, by computing its
primitive central idempotents and the corresponding simple components, in terms of subgroups of G. Basing on the work in <cit.> and <cit.>, a
more precise description of 𝔽_qG, where G is a finite metabelian group, has been given by Bakshi et al. in a series of papers 
<cit.>.

The present paper is a contribution to the work in same series. Recall that a group G is called normally monomial, if every irreducible character of G is  normally monomial, i.e., induced from a linear character of a normal subgroup of G. 
We provide a complete set of primitive central idempotents and the Wedderburn decomposition of 𝔽_qG, when G is a finite normally monomial group. Since metabelian groups are normally monomial <cit.>, this generalises the main result of <cit.>. It may be remarked that normally monomial groups form a substantial class of monomial groups <cit.> and the rational group algebra of this class of groups has been studied <cit.>.  

The main result is given in Section 2, after setting up necessary notation and preliminaries  and is illustrated on a family of metabelian groups. In Section 3, we give applications of the main result on certain p-groups, which include a family of non-metabelian but normally monomial groups of order p^7, p prime, p≥ 5. It may be pointed out that for most of these group algebras, the  package  <cit.> practically fails to compute the
Wedderburn decomposition. Further, we provide the explicit structure of 𝔽_qG, for any group G of order p^n, p prime, n<5. For these groups, the  group Aut(𝔽_qG) of 𝔽_q-automorphisms of 𝔽_qG  has also been computed. 


	

§ NOTATION

	Throughout the paper, G denotes a finite group, and 𝔽_qG denotes the group algebra of G over the field 𝔽_q containing q elements, q relatively prime to the order |G| of G. The notation used are mostly standard and are listed for the ease of reader.

	
	[                                                                    H ≤ ( ) G                                   H  is  a  subgroup  (normal subgroup) of G;                                                                       N_G(H)                                          the  normalizer  of  H in G,  H ≤ G;                                                                        [G:H]                            the  index  of  the  subgroup  H   in   G,  H ≤ G;                                                                          H^g                                                       g^-1Hg,  g ∈ G,  H ≤ G;                                                                    core_G(H) ⋂_g∈ GH^g ,  the  largest  normal  subgroup  of  G   contained in  H,  H ≤ G;                                                                        [h,g]                                                        h^-1g^-1hg,  g, h ∈ G;                                                                           G'                                                         {[h,g]  |  g, h ∈ G};                                                                              ]
			[                                                                              |X|                                                 the cardinality  of  the  set  X;                                                                             ⟨ X⟩                            the   subgroup  generated  by  the  subset  X   of  G;                                                                              ψ^G                      the  character  ψ  of   a  subgroup of  G,  induced  to   G;                                                                              𝔽_q                                                   an  algebraic  closure  of 𝔽_q;                                                                           Irr(G)     the  set  of  all  the  distinct 
		irreducible  characters  of  G  over 𝔽_q;                                                                           ker(χ)                                                { g ∈ G  | χ(g) = χ(1)}, χ∈Irr(G);                                                                           𝔽_q(χ)           the  field  obtained   by  adjoining  χ (g), g ∈ G,  to 𝔽_q , χ∈Irr(G);                                                                  Gal(𝔽_q(χ)/𝔽_q)                                          the Galois  group  of  𝔽_q(χ)  over 𝔽_q;                                                                             e(χ)                                              χ(1)/|G|∑ _g ∈ Gχ(g) g^-1, χ∈Irr(G);                                                                         e_𝔽_q(χ)    ∑_σ ∈ Gal(𝔽_q(χ)/ 𝔽_q) e( σ∘χ),  the  primitive  central  idempotent  of 𝔽_qG;                                                                                                                       associated  to  the  character χ , χ∈Irr(G);                                                                     a| b  (a∤ b)                                     a   divides  (does  not  divide)  b, a,b ≥ 1;                                                                                φ                                                           Euler's  phi  function;                                                                         gcd(k,n)                        the  greatest  common  divisor  of the integers k  and  n;                                                                              S_n                                     the  symmetric  group  on  n   symbols, n≥ 1;                                                                              ℤ_n                                        the  cyclic  group
		 of  order  n, n ≥ 1;                                                                          SL_n(𝔽) the  group  of  n× n   matrices  of  determinant  1,   over  the  field 𝔽, n ≥ 1;                                                                        H_1⋊  H_2                    the  split  extension  of  the  group H_1  by  the  group H_2;                                                                           M_n(𝔽)                         the ring  of  n × n  matrices  over  the  field 𝔽, n ≥ 1;                                                                       M_n(𝔽)^(s)               M_n(𝔽)⊕ M_n(𝔽)⊕...⊕ M_n(𝔽),  the direct  sum  of  s  copies,  s≥ 1;                                                                                  ]
	
	

§ NORMALLY MONOMIAL GROUPS
  
	

 §.§ Strong Shoda pairs and idempotents

	For H K ≤ G, define 
    ε (K, H) :=  K̂,     K = H
    ∏ (Ĥ-L̂),     otherwise,
 where K̂ :=1/|K|∑_k ∈ K k and L runs over the normal subgroups of K which are minimal over the normal subgroups of K containing H properly. Set e(G,K,H) to be the sum of distinct G-conjugates of ε(K,H).  A strong Shoda pair <cit.> of G is a pair (K, H) of subgroups of G with the property that   (i) H  K   N_G(H); 
 (ii) K/H is cyclic and a maximal abelian subgroup of N_G(H)/H;
 (iii) the distinct G-conjugates of ε(K, H) are mutually orthogonal.  Two strong Shoda pairs (K_1, H_1) and (K_2, H_2) are said to be equivalent, ife(G, K_
		1, H_1) = e(G, K_2, H_2) and a complete set of representatives from distinct equivalence classes of strong  Shoda pairs of G is called a complete  irredundant set of strong Shoda  pairs of G.
	

			
	We recall the algorithm to compute a complete irredundant set of strong Shoda pairs of a finite normally monomial group G, as described in <cit.>.
	
	Let 𝒩 be the set of all normal subgroups of G and for N∈𝒩,  let A_N be a normal subgroup of G containing N such that A_N/N is an abelian normal subgroup of maximal order in G/N. Note that the choice of A_N is not unique. However, we need to fix one such A_N. For a fixed A_N, set 

 [                                                                                 𝒟_N:   the  set  of  all  subgroups  D   of A_N  containing   N   such  that core_G(D)=N,;  A_N/D   is   cyclic    and    is  a   maximal  abelian   subgroup   of    N_G(D)/D.;                                                                                 𝒯_N: a  set  of  representatives  of 𝒟_N   under  the  equivalence  relation  defined  by;                                                                                                                                           conjugacy  of  subgroups in  G.;                                                                                 𝒮_N:                                                                {( A_N,D) |  D ∈𝒯_N}. ]
	
	It thus follows that if N ∈𝒩 is such that G/N is abelian, then 
    𝒮_N = {(G, N)},     if G/N  is  cyclic,
    ∅,     otherwise.

	
	
	Observe that every pair (A,D)∈𝒮(G), where 
    𝒮(G):=⋃_N ∈𝒩𝒮_N,
 is a strong Shoda pair of G. It has been proved (<cit.>, Corollary 1) that 𝒮(G)
	is a complete irredundant set of strong Shoda pairs of G, if G is a finite normally monomial group. 

	
	Remark 1.
	A crucial observation in the above algorithm to compute 𝒮(G), for a given finite group G, is that the choice of A_N is irrelevant. For N ∈𝒩, let A_N^(1) be another normal subgroup of G containing N such that A_N^(1)/N is an abelian normal subgroup of maximal order in G/N and let 𝒟_N^(1), 𝒯_N^(1) and 𝒮_N^(1) be defined corresponding to A_N^(1). Then, any pair in 𝒮_N^(1) is equivalent to a pair in 𝒮_N and vice-versa. This is because, if (A_N^(1),D^(1))∈𝒮_N^(1) and ψ is a complex linear character of A_N^(1) with kernel D^(1), then ψ^G is irreducible and hence by (<cit.>, Lemma 1), there exists (A_N,D)∈𝒮_N such that e_ℚ(ψ^G), the primitive central idempotent of the rational group algebra ℚG, associated to ψ^G, is given by e(G,A_N,D). However, in view of (<cit.>, 
Theorem 2.1), e_ℚ(ψ^G)=e(G,A_N^(1),D^(1)). This gives that (A_N,D) is equivalent to (A_N^(1)
	,D^(1)). The reverse conclusion holds similarly.

	
	For a strong Shoda pair (K,H) of G, let 𝒞(K/H) denote the set of q-cyclotomic cosets of (K/H) containing its generators, i.e., if χ is a generator of (K/H), then an element C of 𝒞(K/H) containing χ is the set {χ, χ^q,…, 
	χ^q^o-1}, where n=|K/H| and o=o_n(q), the order of q modulo n. Suppose that N_G(H) acts on 𝒞(K/H) by conjugation, i.e., for g∈ N_G(H), C ∈𝒞(K/H) and χ∈ C,
	we define C^g={χ^g, χ^g^q,…,
	χ^g^q^o-1}, where χ^g(k)=χ(gkg^-1). Let ℛ(K/H) denote the set of distinct orbits of 𝒞(K/H)  and let
	E_G(K/H) be the stabilizer of any C∈𝒞(K/H) under the above action. It is easy to see that 
	|ℛ(K/H)|=ϕ(n)|E_G(K/H)|/|N_G(H)|o_n(q). 
	
	For C∈𝒞(K/H) and χ∈ C, set 
	ε_C(K, H) =1/|K|∑_g ∈ K
	(tr(χ(g)))g^-1, where g denotes the image of g in K/H, ζ_n a primitive n^th 
	root of unity in 𝔽_q and tr:=tr_𝔽_q(ζ_n)/𝔽_q denotes the trace of the field extension 𝔽_q(ζ_n)/𝔽_q. Let e_C(G, K, H) denote the sum of distinct G-conjugates of ε_C(K, H). Broche and Rio <cit.> proved that
	
    e_C(G,K,H)   is  a  primitive  central  idempotent  of 𝔽_qG

	and
	
    𝔽_qGe_C(G,K,H)≅ M_[G:K](𝔽_q^o/[E
    			:K]),

		where E=E_G(K/H) and o=o_[K:H](q).

	
	

 §.§ Main theorem

		The following theorem gives a complete set of primitive central idempotents of finite semisimple group algebra 𝔽_qG, when G is a normally monomial group. Consequently, the complete algebraic structure of 𝔽_qG is obtained.
	
		Let  𝔽_q be a finite field with q elements and let 
		G be a finite  group. Suppose that gcd(q, |G|) = 1. Then,
		
    E:={e_C(G, A, D) | (A, D)∈𝒮(G),  C∈ℛ(A/D)}
 is a complete
		set of primitive central idempotents of 𝔽_qG if, and only if, G is normally monomial.
		
	

In order to prove the above theorem, we need the following lemmas:

(<cit.>, Lemma 1) Let 𝔽_q be a finite field with q elements and let 
	G be a finite  group. Suppose that gcd(q, |G|) = 1. Let ψ be a linear character of a normal subgroup A of G with kernel D. 
	If ψ^G∈(G), then e_𝔽_q (ψ^G)=e_C(G, A, D), for someC∈𝒞(A/D).
	

(<cit.>, Lemma 2.2) Let A be an abelian normal subgroup of maximal order in a group G. If χ is a faithful irreducible normally monomial character of G, then χ is induced from A. 
Proof. By assumption, χ = β^G for some linear character β of a
normal subgroup B of G. As B'⊆core_G(ker (β))=ker (χ)= ⟨ 1⟩, we get that B is abelian and hence χ(1)=[G:B] ≥ [G:A]. Let α be an irreducible constituent of χ_A, where χ_A denotes the restriction of χ to A. By Frobenius reciprocity (<cit.>, Corollary 4.2.2), χ is a constituent of α^G, so that χ(1) ≤α^G(1) = [G : A]. Thus, χ(1) = [G : A] = α^G(1) and χ = α^G. 


Proof of Theorem <ref>. Assume first that E is a complete set of primitive central idempotents of 𝔽_qG. 
	Let χ∈ (G), so that  e_𝔽_q (χ)=e_C(G, A, D), where (A, D)∈𝒮(G) and C∈ℛ(A/D). But then, it follows from Lemma <ref> that
	e_C(G, A, D)= e_𝔽_q (ψ^G), for some linear character 
	ψ of A with kernel D. Hence e_𝔽_q (χ)=e_𝔽_q (ψ^G), which implies that χ = σ∘ψ^G, where 
	σ∈Gal(𝔽_q(ψ^G)/𝔽_q), i.e., χ is induced from a linear character σ∘ψ of a normal subgroup 
	A of G. Thus, G is normally monomial.
	
	Conversely, if G is a normally monomial group, then 
	for χ∈Irr(G), χ = ψ^G, where ψ is a linear character of some normal subgroup K of G 
	with kernel H. Now, ker(χ)= core_G(H)=N(say). Let χ be the corresponding character of G/N. It follows from Lemma <ref> that  
    χ = η^G/N
  for some linear character η of A_N/N, where A_N/N is an abelian normal subgroup of maximal order in G/N. Let (η)= L/N and let η : A_N→𝔽_q be given  by η(g)= η(gN), g∈ A_N. We have, by Eq. (<ref>),  χ = η^G, and therefore, by  Lemma <ref>, it follows that e_𝔽_q(χ)= e_𝔽_q(η^G)=e_C(G, A_N, L),  C∈𝒞(A_N/L). 
	 Now, if D∈𝒯_N is
	a representative of conjugacy class of L, then D=L^x, for some x∈ G and e_C(G, A_N, L)=
	e_C^x(G, A_N, L^x), with (A_N, D)∈𝒮(G).Consequently, e_𝔽_q(χ)∈ E. Moreover, it is easy to check that all elements in E are distinct and hence, E is a complete set of primitive central idempotents of 𝔽_qG.  

	 
	 
	 It thus follows from Theorem <ref> and Eq. (<ref>) that if G is normally monomial, then 
	
    𝔽_qG≅⊕_(A,D)∈𝒮(G)M_[G:A](𝔽_q^o/[E
    				:A])^(|ℛ(A/D)|),

	
	where E=E_G(A/D) and o=o_[A:D](q).
	

 §.§ An illustration

	 We first illustrate the results of Theorem 1 on a family of metabelian groups of order 2sp^2, p an odd prime and s≥ 1. The structure of the rational group algebra of this family has been described in <cit.>. We undertake the case of finite semisimple group algebra for these groups.
	
	Let G be a group generated by a, b, x, y satisfying 
		a^p=b^p=x^s=y^2=1, [a,b]=[x,y]=1,
		x^-1ax=a^r, x^-1bx=b^r, y^-1ay=b, y^-1by=a, where p is an odd prime, r is a positive integer such that gcd(r, p)=1 and  s(>1) is the order o_p(r) of r modulo p. The Wedderburn decomposition of  𝔽_qG is given as follows:

	
		(i) If s is odd, then 

		
			
    𝔽_qG ≅ {[                                     ⊕_d|s𝔽_q^o_d^(2ϕ(d)/o_d)⊕ M_s(𝔽_q^fg/s)^(2(p-1)/fg)⊕ M_2s(𝔽_q^fg/s)^(p^2-p/2fg),                                                                                                                 2∤ f; ⊕_d|s𝔽_q^o_d^(2ϕ(d)/o_d)⊕ 
    			M_s(𝔽_q^fg/s)^(2(p-1)/fg)⊕ M_2s(𝔽_q^fg/2s)^(p-1/fg)⊕ 
    			M_2s(𝔽_q^fg/s)^((p-1)^2/2fg),                                                                                                                  2|f ];
    			.

		
		

			
			
			(ii) If s is even, then 

			
				
    𝔽_qG ≅ {[ ⊕_d|s𝔽_q^o_d^(2ϕ(d)/o_d)⊕ M_s(𝔽_q^fg/s)^(2(p-1)/fg)⊕ 
    				M_2s(𝔽_q^fg/s)^((p-1)^2/2fg)⊕ 
    				M_s(𝔽_q^2fg'/s)^((p-1)/fg'),                                                                                                                      2∤ f;                              ⊕_d|s𝔽_q^o_d^(2ϕ(d)/o_d)⊕ 
    				M_s(𝔽_q^fg/s)^(4(p-1)/fg)⊕ M_2s(𝔽_q^fg/s)^
    				((p-1)^2/2fg),                                                                                                                       2|f ],
    				.

				
			
		

		
		

		
		where
		o_d=o_d(q), f=o_p(q)=o_2p(q), g=gcd{d|s : s/d|f} and g'=gcd{d | s/2 : s/2d , s/2d | f}.
		
	
	
	
	Proof. The distinct normal subgroups of G (<cit.>, Lemma 2) are  N_0:=⟨ 1 ⟩, N_1:=⟨ ab⟩,  N_2:=⟨ ab^-1⟩,  N_3:=⟨ ab^-1, y⟩, G_d:= ⟨ a, b, x^d⟩,  H_d:= ⟨ a, b, x^d, y⟩, d|s, if s is odd; and, in addition, N_4:= ⟨ ab, x^s/2y⟩,  K_d:= ⟨ a, b, x^d/2y ⟩, with d even and d|s, if s is even. Further, for each normal subgroup N of G, the corresponding set 𝒮_N of strong Shoda pairs of G (<cit.>, Lemma 3) is as follows:
	
	
	(i) 𝒮_N_0 ={(⟨ a,b ⟩, ⟨  b ⟩)}∪{( ⟨ a, b ⟩, ⟨ ba^λ ^i⟩) |  1≤ i ≤p-3/2}, where λ is generator of the multiplicative group  of reduced residue classes modulo p.


	(ii) 𝒮_N_1 = {(G_s,N_1)},     s   odd

{( K_s,N_1)},    s  even. 

 
	(iii) 𝒮_N_i =  {(H_s,N_i)},   i =2, 3. 

(iv)
	𝒮_N_4 =  {( K_s,N_4)}.  


	(v) 𝒮_G_d = {( G,G_d)},     d   odd
∅  ,    d  even. 

(vi)  𝒮_H_d= { ( G,H_d)}, d|s. 


	(vii)
	𝒮_K_d= { (G,K_d)}, d|s, d even. 

	
	We tabulate the computations of parameters involved in applying Theorem <ref>.

	
	s odd
			
    =.5pt1.4(A_N, D)∈𝒮(G)   E_G(A_N,D)   o(A_N,D)   |ℛ(A_N/D)|
    (G,G_d), d|s   G   o_2d(q)=o_d(q)   ϕ(d)/o_d(q)
    (G,H_d),  d|s   G   o_d(q)   ϕ(d)/o_d(q)
    (G_s,N_1)   {[ <a,b,x^g,y>,          2|f;   <a,b,x^g>,         2∤ f ].   {[ fg/2s,    2|f;  fg/s,   2∤ f ].   {[  p-1/fg,      2|f; p-1/2fg,     2∤ f ].
    (H_s,N_2)   <a,b,x^g,y>   fg/s   p-1/fg
    (H_s,N_3)   <a,b,x^g,y>   fg/s   p-1/fg
    (<a,b>,<b>)   <a,b,x^g>   fg/s   p-1/fg
    (<a,b>,<ba^λ^i>)   2*<a,b,x^g>   2*fg/s   2*p-1/fg
    1≤ i ≤p-3/2

	
	

	s even
		
    =.5pt1.4(A_N,D)   E_G(A_N,D)   o(A_N,D)   |ℛ(A_N/D)|
    (G,G_d), d|s d   G   o_2d(q)=o_d(q)   ϕ(d)/o_d(q)
    (G,K_d), d|s, d   G   o_d(q)   ϕ(d)/o_d(q)
    (K_s,N_1)   {[ <a,b,x^g,y>,          2|f; <a,b,x^g'y>,         2∤ f ].   {[   fg/s,     2|f; 2fg'/s,    2∤ f ].   {[   p-1/fg,       2|f; p-1/2fg',      2∤ f ].
    (K_s,N_4)   {[ <a,b,x^g,y>,          2|f; <a,b,x^g'y>,         2∤ f ].   {[   fg/s,     2|f; 2fg'/s,    2∤ f ].   {[   p-1/fg,       2|f; p-1/2fg',      2∤ f ].
    (H_s,N_2,)   <a,b,x^g,y>   fg/s   p-1/fg
    (H_s,N_3)   <a,b,x^g,y>   fg/s   p-1/fg
    (<a,b>,<b>)   <a,b,x^g>   fg/s   p-1/fg
    (<a,b>,<ba^λ^i>)   2*<a,b,x^g>   2*fg/s   2*p-1/fg
    1≤ i ≤p-3/2

	
	
	

	

	
Consequently, Theorem <ref> follows using Eq. (<ref>). 


§ APPLICATIONS ON P-GROUPS
 
	

 §.§ A family of normally monomial groups of order p^7

	We now illustrate the main result of this paper on a normally monomial group of order p^7,  p a prime, p≥ 5, which is not metabelian.
	
	Let p≥ 5 be a prime and let G be the group generated by a, b, c, d, s, r with the following defining relations:
		  a^p=b^p=c^p=d^p=s^p=r^p=1,

			[a,c]=[b,c]=[a,d]=[b,d]=1, [a,b]=[c,d],

			s^-1as=a,  s^-1bs=bc,  s^-1cs=acd,  s^-1ds= ad, 

			[s,r]=[a,b], [r,a]=[r,b]=[r,c]=[r,d]=1.		
			
			
		Then, 
	
			
    𝔽_qG ≅ 𝔽_q⊕𝔽_q^f^(p^3-1/f)⊕ M_p( 𝔽_q^f)^(p^4-p/f)⊕ M_p^3( 𝔽_q^f)^(p-1/f),

			 where f=o_p(q).

	
	
	Proof. Let G be as in statement of the theorem.  How <cit.> proved that G is a normally monomial group, which is not metabelian. We find 𝒮(G) to write the Wedderburn decomposition of 𝔽_qG.
	
	First of all, we compute the set 𝒩 of normal subgroups of G. For this,
	we begin by observing that if ⟨ 1⟩≠ N∈𝒩, then 
    [a,b]∈ N.
Let 1≠ g=a^αb^βc^γd^δr^ρs^ζ[a,b]^η∈ N, where 0≤α, β, γ, δ, ρ, ζ, η < p. Since r^-1gr=g[a,b]^ζ∈ N, we obtain that ζ=0, if [a,b]∉N. Therefore, g=a^αb^βc^γd^δr^ρ[a,b]^η and b^-1gb=g[a,b]^α now yields α=0, if [a,b]∉N. Proceeding similarly, we obtain that if [a,b]∉N, then g=1, which contradicts the assumption and proves (<ref>).
	
	We next check that if a∉N, then 
    N=⟨[a,b]⟩ or  ⟨[a,b], ra^i⟩ ,  where  0≤ i<p.
 
	In view of (<ref>), we already have that either N=⟨[a,b]⟩ or there exists an element g=a^αb^βc^γd^δr^ρs^ζ∈ N∖⟨ [a,b]⟩, where 0≤α, β, γ, δ, ρ, ζ < p. Observe that d^-1gd=g[a,b]^γa^-ζ∈ N implies ζ=0, if a∉N. Continuing in this manner, repeated conjugacy by s implies δ=γ=β=0, if a∉N and g=a^αr^ρ, ρ≠ 0.  Thus, (<ref>) follows.
	
	Next, assume  a∈ N, so that ⟨ a, b^-1ab⟩ (∈𝒩)≤ N, i.e., we look for normal subgroups N of G containing ⟨ a, b^-1ab⟩, which are in one-one correspondence with the normal subgroups N of G:=G/⟨ a, b^-1ab⟩. Proceeding as previously, we see that if 1≠g=b^βc^γd^δr^ρs^ζ∈N, for some 0≤β, γ, δ, ρ, ζ < p, then either d∈N or g=d^δr^ρ, ρ≠ 0, and in the later case, 
    N=⟨ a, b^-1ab⟩ or  ⟨ a, b^-1ab, rd^i⟩ ,   where  0≤ i<p.
 
	Now, if d∈N, then N is a normal subgroup of G, which contains the normal subgroup ⟨d⟩ of G. Hence, N corresponds to a normal subgroup N of G:=G/⟨d⟩. Continuing this process, the set of normal subgroups of G, is obtained as 
	
    𝒩={G,N_0,N_1,N_2,N_3,N_4,N_5,N_6,G^(1)_i,G^(2)_i,G^(3)_i,G^(4)_i,G^(5)_i,H_ij,K_ij |  0≤ i,j < p},
 
	where  N_0:=⟨ 1 ⟩;  N_1:=⟨ a,b,c,d,r ⟩; N_2:=⟨ a,b,c,d ⟩; N_3:=⟨ a,c,d ⟩;
N_4:=⟨ a,b^-1ab, d ⟩; N_5:=⟨ a,b^-1ab ⟩; N_6:=⟨ [a,b] ⟩;
G^(1)_i:=⟨ a,b,c,d,sr^i⟩; G^(2)_i:=⟨ a,c,d,rb^i⟩; G^(3)_i:=⟨ a,b^-1ab,d,rc^i⟩;
G^(4)_i:=⟨ a,b^-1ab,rd^i⟩; G^(5)_i:=⟨ [a,b],ra^i⟩;
H_ij:=⟨ a,c,d,rb^i,sb^j⟩; and K_ij:=⟨ a,c,d,sr^ib^j⟩, for 0≤ i,j < p.
	
	Note that if N=G,N_1,H_ij,G^(1)_i,N_2,K_ij,G^(2)_i or N_3, then G/N is abelian and hence, by Eq. (<ref>), we have
	
	
    𝒮_N = {(G, N)},     if  N∈𝒩_1:={G,N_1,H_ij,G^(1)_i},
    ∅,     if N∈𝒩_2:={N_2,K_ij,G^(2)_i,N_3}.

	Further, if N≠ N_0 is such that G/N is not abelian, then N_1/N is abelian.Consequently,
	
	
	
    A_N=N_1,   if N∈𝒩_3:={G^(3)_i,N_4,G^(4)_i,N_5,G^(5)_i,N_6}.

	Also, it is easy to observe that 
    A_N=G^(3)_0,   if  N∈𝒩_4:={N_0}
 serves the purpose. 

	
	Since 𝒩=𝒩_1 ∪ 𝒩_2 ∪ 𝒩_3 ∪ 𝒩_4, in view of (<ref>)-(<ref>), we need to find 𝒯_N,  when N∈𝒩_3 ∪ 𝒩_4. We first take up the case when N∈𝒩_4, i.e., N=N_0=⟨ 1⟩. In this case, since A_N_0=G^(3)_0 is an abelian p-group, the set 𝒞_0 of subgroups of  A_N_0, which give cyclic quotient can be computed using the algorithm given in Section 5 of <cit.> and is as follows:
	
    𝒞_0={H^(0),H^(1),H_β,H_αβ,H_αβδ | 0≤α,β,δ<p},

	where 
	H^(0):=⟨ a, d, r , b^-1ab⟩, H^(1):=⟨ a, d, r  ⟩, H_β:=⟨ d, r , b^-1aba^β⟩, H_αβ:=⟨ ad^α, r , b^-1abd^β⟩, H_αβδ:=⟨ ar^α, b^-1abr^β, dr^δ⟩, for 0≤α, β, δ<p.  In view of (<ref>), we note that for H∈𝒞_0, core_G(H)=N_0=⟨ 1 ⟩, if, and only if, [a,b]∉H. This yields,
	
    𝒟_N_0={H^(1)} ∪ {H_β | β≠ -1}∪{H_αβ, H_αβδ | α≠β, 0≤α,β,δ<p}.

	It turns out that all subgroups of 𝒟_N_0 are G-conjugates and hence 
	
    𝒯_N_0={H^(1)}.

	For ease of the reader, we list the conjugating element g∈ G such that H^g=H^(1), for every H∈𝒟_N_0.

	
		
  * H_β^x=H^(1), with x=b^ν, where ν(β+1)≡ -1
		 (mod  p), β≠ -1.
		
  * H_αβ^y=H^(1), with y=(b^αc)^λ, where λ(β-α)≡ 1 (mod  p), α≠β.
		
  * H_αβδ^z=H^(1), with z=(b^αc^-δ)^λs^λ, where λ(β-α)≡ 1 (mod  p), α≠β.	
		
	
	
	Finally, we proceed to find 𝒮_N, when N∈𝒩_3. For this, we find the set 𝒞_1 of subgroups of A:=N_1=⟨ a, b , c, d , r⟩ ( see (<ref>)) which give cyclic quotient, so that 
	
    𝒟_N:={D∈𝒞_1 | core_G(D)=N}.

	Let D∈𝒞_1. Then, A/D being cyclic ensures [a,b]∈ D. Moreover, any element g=a^αb^βc^γd^δr^ρ∈A:=A/D, where 0≤α, β, γ, δ, ρ< p, is such that g^p=1, so that A=⟨1⟩ or a cyclic group of order p. If a∉D, then a∈A yields A=⟨a⟩. Also, b∈A implies ba^β∈ D, for some 0≤β<p. Similarly, ca^γ, ra^ρ, da^δ∈ D, for some 0≤γ, ρ, δ<p. This, along with (<ref>) and order considerations yield D=D_ρδγβ:=⟨ [a,b],ra^ρ,da^δ,ca^γ,ba^β⟩,  for some 0 ≤ρ,δ,γ,β<p. Assuming next, a∈ D and b∉D gives D=D_ρδγ:=⟨ a,[a,b],rb^ρ,db^δ
	,cb^γ⟩, 0 ≤ρ,δ,γ<p. Iterating in the above manner, we obtain
	
    𝒞_1:={D^(0),D^(1),D_ρ,D_ρδ,D_ρδγ,D_ρδγβ|  0≤ρ,δ,γ,β<p},

	where D^(0):=A, D^(1):=⟨ a,b,c,d ⟩,  D_ρ:=⟨ a,b,c,rd^ρ⟩, D_ρδ:=⟨ a,b,dc^δ,rc^ρ⟩, and  D_ρδγ, D_ρδγβ are as defined above, 0 ≤ρ,δ,γ,β<p. Since core_G(D) is the largest element of 𝒩,  contained in D, in view of (<ref>), (<ref>) & (<ref>),  we obtain:
	
	
		
  * 𝒟_N=∅, if N=N_4,N_5 or N_6.
		
  * 𝒟_N={ D_i0}∪{ D_ρ0γ | ρ≡-γ i (mod p),  0< γ<p},  if  N=G^(3)_i,  0≤ i < p.
		
		
  * 𝒟_N={ D_i}∪{D_ρδ,  D_ρδγ |  ρ≡-δ i (mod  p), 0<δ<p, 0≤γ<p},  if  N=G^(4)_i,   0≤ i < p.
		
  * 𝒟_N={D_iδγβ | 0≤δ,γ,β<p}, if N=G^(5)_i,  0≤ i < p.
	
	We now omit the details for the remaining part and enlist the set  𝒯_N, whenN=G^(3)_i, G^(4)_i,  G^(5)_i, 0≤ i < p.
	
	
		
  * 𝒯_N={ D_i0},  if  N=G^(3)_i,   0≤ i < p.
		
		
  * 𝒯_N={ D_i}∪{ D_iδ0 |  0<δ<p},  if  N=G^(4)_i,  0≤ i < p.
		
		
  * 𝒯_N={D_i0γβ, 0≤γ, β<p}, if N=G^(5)_i, 0≤ i < p.
	
	In view of (<ref>) and Eq. (<ref>), the information gathered above yields the complete set of strong Shoda pairs of G, and their respective simple components, as tabulated below:
	
	
		
    =6.5pt1.41Normal Subgroup, N   A_N   𝒮_N   |ℛ(A_N/D)|   ⊕_(A,D)∈𝒮_N𝔽_qGe_𝒞(G,A_N,D)
    G   G   {(G, G)}   1    𝔽_q
    N_1=⟨ a, b, c, d, r⟩   G   {(G, N_1)}   p-1/f   𝔽_q^f
    H_i,j=⟨ a, c, d, rb^i, sb^j⟩   2*G   2*{(G, H_ij)}   2*p-1/f   𝔽_q^f
    0≤ i, j < p            0≤ i, j < p
    G_i^(1)=⟨ a,b,c,d,sr^i⟩   2*G   2*{(G, G_i^(1))}   2*p-1/f   𝔽_q^f
    0≤ i < p            0≤ i < p
    N_2= ⟨ a,c,d,b⟩   G   ∅    -    -
    K_i,j=⟨ a, c, d, sr^ib^j⟩,      2*G   2*∅   2*-   2*-
    0≤ i, j < p            
    G_i^(2)= ⟨ a, c, d, rb^i⟩ ,    2*G   2*∅   2*-   2*-
    0≤ i < p            
    N_3=⟨ a,c,d⟩   G   ∅   -    - 
    G_i^(3)= ⟨ a, d, b^-1ab, rc^i⟩,   2*N_1   2*{(N_1, ⟨ a, b, d,  rc^i⟩)}   2*p-1/f   M_p(𝔽_q^f)
    0≤ i < p            0≤ i < p
    N_4=⟨ a,b^-1ab, d⟩   N_1   ∅   -    - 
    G_i^(4)= ⟨ a,  b^-1ab, rd^i⟩,   2*N_1   {(N_1, ⟨ a,b,c,rd^i⟩)} ∪   2*p-1/f   M_p(𝔽_q^f)^(p)
    0≤ i < p      {(N_1, ⟨ a,b^-1ab,c,db^δ,rd^i⟩)  |  1 ≤δ < p  }      0≤ i < p
    N_5=⟨ a, b^-1ab⟩   N_1   ∅    -   - 
    G_i^(5)= ⟨ [a,b], ra^i⟩,   2*N_1   2*{(N_1, ⟨ [a,b],d,ca^γ,ba^β,ra^i⟩)  |  0 ≤β,γ  < p  }   2*p-1/f   M_p(𝔽_q^f)^(p^2)
    0≤ i < p            0≤ i < p
    N_6=⟨ [a,b]⟩   N_1   ∅   -   -                
    N_0=⟨ 1 ⟩   G_0^(3)   {(⟨ a, d, r, b^-1ab⟩, ⟨ a, d, r⟩)}   p-1/f   M_p^3(𝔽_q^f)

	 

	Hence, the desired result is obtained. 

	
	Remarks. Let G be as in statement of Theorem <ref>. Then, by the irredundant set of strong Shoda pairs of G, obtained in Theorem <ref> and (<cit.>, Corollary 1), it follows that 
	
    ℚG≅ℚ⊕ℚ(ζ_p)^(1+p+p^2)⊕ M_p(ℚ(ζ_p))^(p(1+p+p^2))⊕ M_p^3(ℚ(ζ_p)).

	
	A complete irredundant set of strong Shoda pairs of G for the case when p=5 has been computed in <cit.>, using  <cit.>, to find the Wedderburn Decomposition of the rational group algebra ℚG. However, it may be pointed out that  package  <cit.> fails to compute the Wedderburn decomposition of ℚG or 𝔽_qG, for any p≥ 5. In fact, though theoretically,  <cit.> could handle the calculation of the Wedderburn decomposition of group algebras of groups of arbitrary size but in practice, if the order of the group is greater than 5000 then the program may crash.  features  the function  that determines a complete irredundant set of strong Shoda pairs of G and
	the function  that computes the set of primitive central idempotents of semisimple group algebra 𝔽G. For the case when 𝔽=ℚ, these functions are based on the search algorithms provided by Olivieri and del Río <cit.>. Based on the work in <cit.>, alternative and more efficient algorithms have been given in <cit.>.  Analogously, in view of Theorem <ref>,  improved algorithms can be written and implemented in , for the case when 𝔽 is a finite field.
	
	
	

 §.§ Groups of order p^n,  p prime, n<5

	The groups of order p^n, p prime, n<5 are metabelian and hence normallymonomial. Therefore, by applying Theorem <ref>, we now give the explicit structure of 𝔽_qG, and its automorphism group when G is a group of order p^n, p prime, n<5. 
	The Wedderburn decomposition of 𝔽_qG, 
	when G is abelian is well known <cit.> and the automorphism group in this case can be computed as in (<cit.>, Theorem 4). We therefore restrict to the case when G is non-abelian.
	

  §.§.§ Groups of order p^3

	p=2
	
	If p=2, then G is either Q_8, the quaternion group of order 8 or D_8, the dihedral group of order 8. In this case, 
	the set of primitive central idempotents and the Wedderburn decomposition of 𝔽_qG can be read from (<cit.>, Examples 4.3 & 4.4).

	
	In fact,
	
    𝔽_qQ_8≅𝔽_qD_8≅𝔽_q^(4)⊕ M_2(𝔽_q),

	and it follows from (<cit.>, Theorem 5) that
	
    Aut(𝔽_qQ_8)≅Aut(𝔽_qD_8)≅ S_4⊕ SL_2(𝔽_q).


	
	p≠ 2

	
	If p ≠ 2, then G is isomorphic to either ⟨ a, b  |  a^p^2=b^p=1,  b^-1ab= a^1+p⟩ or ⟨ a,b,c  |  a^p=b^p=c^p=1, ab=bac, ac=ca, bc=cb ⟩ (<cit.>, 4.4). The strong Shoda pairs of each of these groups, found in (<cit.>, Theorems 3 & 4) along with Theorem <ref>, Eq. (<ref>) and  (<cit.>, Theorem 5), yield the following:
	
	
	
		Let G be a non-abelian group of order p^3, where p is an odd prime. Then, 
		
    𝔽_qG≅𝔽_q⊕𝔽_q^f^((1+p)e)⊕ M_p(𝔽_q^f)^(e),

		and 
		
    Aut(𝔽_qG)≅ 
    		(ℤ_f^((1+p)e)⋊ S_(1+p)e)⊕((SL_p(𝔽_q^f)⋊ℤ_f)^(e)⋊ S_e), 
    		    f≠ 1
    
    		S_(2+p)e⊕( SL_p(𝔽_q)^(e)⋊ S_e),     f=1 
    ,


		where f=o_p(q) and e=p-1/f.
		
		
	
	
	
	
	
	
	
	
	
	
	

  §.§.§ Groups of order p^4

	
	p=2

	
	If p=2, then up to isomorphism, there are 9 non-isomorphic groups of order 2^4 (<cit.>, 118) as listed below:
	
	
		
		
  * ℋ_1:=⟨ a, b : a^8 =b^2 =1, ba=a^5b  ⟩;
		
  * ℋ_2:=⟨ a, b, c : a^4 =b^2=c^2 =1, cb=a^2bc, ab=ba,  ac=ca  ⟩;
		
  * ℋ_3:=⟨ a, b : a^4 =b^4 =1, ba=a^3b  ⟩;
		
  * ℋ_4:=⟨ a, b, c : a^4 =b^2 =c^2=1, ca=a^3c, ba=ab, cb=bc  ⟩;
		
  * ℋ_5:=⟨ a, b, c : a^4 =b^2 =c^2=1, ca=abc, ba=ab, cb=bc   ⟩;
		
  * ℋ_6:=⟨ a, b, c : a^4 =b^4 =c^2=1, ba=a^3b, ca=ac, cb=bc,  a^2=b^2⟩;
		
  * ℋ_7:=⟨ a, b : a^8 =b^2 =1, ba=a^7b  ⟩;
		
  * ℋ_8:=⟨ a, b : a^8 =b^2 =1, ba=a^3b  ⟩;
		
  * ℋ_9:=⟨ a, b : a^8 =b^4 =1, ba=a^7b, a^4=b^2⟩.
	
	
	
	
	 The Wedderburn decomposition of 𝔽_qℋ_i, 1≤ i ≤ 9, is as follows:
		
		
			
  (i) 𝔽_qℋ_1≅𝔽_q^(8)⊕M_2(𝔽_q)^(2)    q≡ 1,5 (mod 8)
𝔽_q^(4)⊕𝔽_q^2^(2)⊕ M_2(𝔽_q^2)     q≡ 3,7 (mod 8);
			
  (ii) 𝔽_qℋ_2≅𝔽_q^(8)⊕M_2(𝔽_q)^(2)    q≡ 1 (mod 4)
𝔽_q^(8)⊕ M_2(𝔽_q^2)     q≡ 3 (mod 4);
			
  (iii) 𝔽_qℋ_3≅𝔽_q^(8)⊕M_2(𝔽_q)^(2)    q≡ 1 (mod 4)
𝔽_q^(4)⊕𝔽_q^2^(2)⊕M_2(𝔽_q)^(2)    q≡ 3 (mod 4);
			
			
  (iv) 𝔽_qℋ_4≅𝔽_q^(8)⊕M_2(𝔽_q)^(2);
			
			
  (v) 𝔽_qℋ_5≅𝔽_q^(8)⊕M_2(𝔽_q)^(2)    q≡ 1 (mod 4)
𝔽_q^(4)⊕𝔽_q^2^(2)⊕M_2(𝔽_q)^(2)    q≡ 3 (mod 4);
			
			
  (vi) 𝔽_qℋ_6≅𝔽_q^(8)⊕M_2(𝔽_q)^(2);
			
			
			
  (vii) 𝔽_qℋ_7≅𝔽_q^(4)⊕M_2(𝔽_q)^(3)    q≡ 1,7 (mod 8)
𝔽_q^(4)⊕ M_2(𝔽_q^2) ⊕ M_2(𝔽_q)     q≡ 3,5 (mod 8);
			
			
			
  (viii) 𝔽_qℋ_8≅𝔽_q^(4)⊕M_2(𝔽_q)^(3)    q≡ 1,3 (mod 8)
𝔽_q^(4)⊕ M_2(𝔽_q^2) ⊕ M_2(𝔽_q)     q≡ 5,7 (mod 8);
			
			
			
  (ix) 𝔽_qℋ_9≅𝔽_q^(4)⊕M_2(𝔽_q)^(3)    q≡ 1,7 (mod 8)
𝔽_q^(4)⊕ M_2(𝔽_q^2) ⊕ M_2(𝔽_q)     q≡ 3,5 (mod 8).
		
		
		
	
	
	
	
	
	
	Proof.
	(i) Define N_0:= ⟨ 1 ⟩, N_1:= ⟨ a^4⟩, N_2:= ⟨ a^2⟩, N_3:= ⟨ a ⟩, H_i:= ⟨ a^4, a^2ib ⟩, K_j:= ⟨ a^2, a^jb ⟩ where 0 ≤ i,j ≤ 1. Observe that these subgroups are normal in ℋ_1. Using Eq. (<ref>), we have 𝒮_N_1= 𝒮_N_2= ϕ, 𝒮_N_3={(ℋ_1,N_3 )}, 𝒮_H_i={(ℋ_1,H_i)}, 𝒮_K_j={(ℋ_1,K_j)}, 0 ≤ i,j ≤ 1. In order to find 𝒮_N_0, we see that ⟨ a ⟩ is a maximal abelian subgroup of ℋ_1. Further, the only subgroup D of ⟨ a ⟩ satisfying core_G(D)=⟨ 1⟩ is D= ⟨1 ⟩. This gives 𝒮_N_0={(⟨ a ⟩, ⟨ 1 ⟩)}.
	Define 
    𝒩_1= {⟨ 1 ⟩, ⟨ a^4⟩, ⟨ a^2⟩, ⟨ a ⟩, ⟨ a,b⟩} ∪ {⟨ a^4, a^2ib ⟩, ⟨ a^2, a^jb ⟩ | 0 ≤ i,j ≤ 1}.
 
	
It follows from Eqs. (<ref>)-(<ref>) that ⊕_N∈𝒩_1⊕_(A_N,D)∈𝒮_N⊕_C∈ℛ(A_N/D)𝔽_qGe_C(G,A_N,D), is a direct summand of 𝔽_qG and has same 𝔽_q-dimension as 𝔽_qG. This yields that if 𝒩 is the set of all normal subgroups of ℋ_1, then
	𝒮_N= ϕ, if N∉𝒩_1, i.e., 𝒮(ℋ_1)=⋃_N∈𝒩_1𝒮_N and
	
	
    𝔽_qℋ_1≅𝔽_q^(8)⊕M_2(𝔽_q)^(2)    q≡ 1,5 (mod 8)
    𝔽_q^(4)⊕𝔽_q^2^(2)⊕ M_2(𝔽_q^2)     q≡ 3,7 (mod 8)
    .
 
	(ii)-(ix) For 2 ≤ i ≤ 9, consider the following set 𝒩_i of normal subgroups of ℋ_i:
		
		
    [                                                                                        𝒩_2 =                                            {⟨ 1 ⟩, ⟨ a^2⟩,  ⟨ a^2,b ⟩, ⟨ a,b ⟩, ⟨ a,b,c ⟩} ∪;                 {⟨ a^2, b^ic ⟩, ⟨ a, b^ic ⟩, ⟨ ab^ic^j⟩, ⟨ a^2, a^ib, a^jc⟩ |  0≤ i,j ≤ 1 };;                                                                                        𝒩_3 =                                                  {⟨ 1 ⟩, ⟨ a^2⟩, ⟨ a ⟩, ⟨ a,b^2⟩, ⟨ a,b ⟩} ∪;                                    { ⟨ a^2ib^2⟩, ⟨ a^2, a^ib ⟩, ⟨ a^2,a^ib^2⟩ |  0≤ i ≤ 1 };;                                                                                        𝒩_4 =                                             {⟨ 1 ⟩, ⟨ a^2⟩, ⟨ a^2,b ⟩, ⟨ a,b ⟩, ⟨ a,b,c ⟩} ∪; {⟨ a^2ib⟩, ⟨ a^2, b^ic ⟩, ⟨ a, b^ic⟩, ⟨ a^2, ab^ic^j⟩, ⟨  a^2,a^ib, a^jc ⟩ |  0≤ i,j ≤ 1 ⟩};;                                                                                        𝒩_5 =    {⟨ 1 ⟩, ⟨ b ⟩, ⟨ a^2⟩, ⟨ a^2b ⟩, ⟨ a^2,b ⟩,  ⟨ a, b ⟩, ⟨ b,ac⟩, ⟨ a^2,b,c ⟩, ⟨ a,b,c ⟩} ∪;                                                                 {⟨ b,a^2ic ⟩ |  0≤ i ≤ 1 ⟩};;                                                                                        𝒩_6 =                                             {⟨ 1 ⟩, ⟨ a^2⟩, ⟨ a^2,c ⟩, ⟨ a,b ⟩, ⟨ a,b,c ⟩} ∪;            {⟨ a^ic⟩, ⟨ bc^i⟩, ⟨ ab^ic^j⟩, ⟨ a, b^ic ⟩, ⟨  a^2,a^ib, a^jc ⟩ |  0≤ i,j ≤ 1 ⟩};;                                                                                        𝒩_7 =                       {⟨ 1 ⟩, ⟨ a^4⟩, ⟨ a^2⟩, ⟨ a ⟩, ⟨ a,b ⟩} ∪ {⟨ a^2,a^ib ⟩ |  0≤ i ≤ 1 };;                                                                                        𝒩_8 =                       {⟨ 1 ⟩, ⟨ a^4⟩, ⟨ a^2⟩, ⟨ a ⟩, ⟨ a,b ⟩} ∪ {⟨ a^2,a^ib ⟩ |  0≤ i ≤ 1 };;                                                                                        𝒩_9 =                       {⟨ 1 ⟩, ⟨ a^4⟩, ⟨ a^2⟩, ⟨ a ⟩, ⟨ a,b ⟩} ∪ {⟨ a^2,a^ib ⟩ |  0≤ i ≤ 1 }.;                                                                                              ]

	
	
	
	Proceeding as in (i), we get the following complete and irredundant set of strong Shoda pairs of ℋ_i, 2 ≤ i ≤ 9, which yield the desired result. 

	
	
	[                                                                         (ii) 𝒮(ℋ_2)                                                                                   =                              {(ℋ_2,  ℋ_2),  (⟨ a, b⟩,  ⟨ b ⟩),   (ℋ_2, ⟨ a, b ⟩)} ∪;                                                                                                                                                                                           {(ℋ_2, ⟨ a,b^ic⟩),   (ℋ_2, ⟨ a^2,a^ib, a^jc⟩) |  0 ≤ i,j ≤ 1 }
	;;                                                                        (iii) 𝒮(ℋ_3)                                                                                   =               {(ℋ_3,  ℋ_3),  (ℋ_3, ⟨ a,b^2⟩),  (ℋ_3, ⟨ a⟩),  (ℋ_3, ⟨ a^2, ab^2⟩)} ∪;                                                                                                                                                                                                {(⟨ a,b^2⟩, ⟨ a^2ib^2⟩),  (ℋ_3, ⟨ a^2, a^ib⟩)  | 0≤ i≤ 1}
	;;                                                                         (iv) 𝒮(ℋ_4)                                                                                   =                                                   {(ℋ_4,  ℋ_4),  (ℋ_4, ⟨ a, b ⟩)} ∪;                                                                                                                                                                         {(⟨ a, b ⟩, ⟨ a^2ib⟩),  (ℋ_4, ⟨ a,b^ic⟩),  (ℋ_4, ⟨ a^2,a^ib, a^jc⟩) | 0≤ i,j≤ 1}
	;;                                                                          (v) 𝒮(ℋ_5)                                                                                   =             {(ℋ_5,  ℋ_5),  (ℋ_5, ⟨ a,b⟩),  (ℋ_5, ⟨ a^2, b,c ⟩),  (ℋ_5,⟨ b,ac⟩)
	} ∪;                                                                                                                                                                                 {(⟨ a, b ⟩, ⟨ a ⟩),  (⟨ a, b ⟩, ⟨ a^2b⟩)} ∪{(ℋ_5, ⟨ b,a^2ic⟩) | 0 ≤ i ≤ 1};;                                                                         (vi) 𝒮(ℋ_6)                                                                                   =                                                    {(ℋ_6, ℋ_6),  (ℋ_6, ⟨ a, b ⟩)} ∪;                                                                                                                                                                         {(⟨ a, c ⟩, ⟨ a^ic⟩),  (ℋ_6, ⟨ a,b^ic⟩ ),  (ℋ_6, ⟨ a^2,a^ib,a^jc⟩ ) | 0 ≤ i,j ≤ 1};;                                                                        (vii) 𝒮(ℋ_7)                                                                                   =                                                       {(ℋ_7,  ℋ_7),  (ℋ_7, ⟨ a⟩)} ∪;                                                                                                                                                                                                   {(⟨ a⟩,  ⟨  a^4i⟩),  (ℋ_7, ⟨  a^2,a^ib ⟩) | 0 ≤ i ≤ 1}
	;;                                                                       (viii) 𝒮(ℋ_8)                                                                                   =                                                       {(ℋ_8,  ℋ_8),  (ℋ_8, ⟨ a⟩)} ∪;                                                                                                                                                                                                   {(⟨ a⟩,  ⟨  a^4i⟩),  (ℋ_8, ⟨  a^2,a^ib ⟩) | 0 ≤ i ≤ 1}
	;;                                                                                     ]
	[                                             (ix) 𝒮(ℋ_9)                                                       =                           {(ℋ_9,  ℋ_9),  (ℋ_9, ⟨ a⟩)} ∪;                                                                                                                 {(⟨ a⟩,  ⟨  a^4i⟩),  (ℋ_9, ⟨  a^2,a^ib ⟩) | 0 ≤ i ≤ 1}.;                                                         ]

	
		The automorphism group of 𝔽_qℋ_i, 1≤ i ≤ 9, is as follows:
		
  * Aut(𝔽_qℋ_1 )≅
			S_8⊕ (SL_2(𝔽_q)^(2)⋊ S_2),     
			q≡ 1,5 (mod 8)

			S_4⊕ (ℤ_2^(2)⋊ S_2) ⊕ (SL_2(𝔽_q^2) ⋊ℤ_2),    
			q≡ 3,7 (mod 8)
;
			
  * Aut(𝔽_qℋ_2 )≅
			S_8⊕ (SL_2(𝔽_q)^(2)⋊ S_2),     q≡ 1 (mod 4)

			S_8⊕ (SL_2(𝔽_q^2) ⋊ℤ_2),     q≡3 (mod 4)
;
			
  * Aut(𝔽_qℋ_3 )≅
			S_8⊕ (SL_2(𝔽_q)^(2)⋊ S_2),     q≡ 1 (mod 4)

			S_4⊕ (ℤ_2^(2)⋊ S_2) ⊕ 
			(SL_2(𝔽_q)^(2)⋊ S_2),    q≡ 3 (mod 4)
;
			
  * Aut(𝔽_qℋ_4 )≅ 
			S_8⊕ (SL_2(𝔽_q)^(2)⋊ S_2) 
			;
			
  * Aut(𝔽_qℋ_5 )≅
			S_8⊕ (SL_2(𝔽_q)^(2)⋊ S_2),     q≡ 1 (mod 4)

			S_4⊕ (ℤ_2^(2)⋊ S_2) 
			⊕ (SL_2(𝔽_q)^(2)⋊ S_2),    q≡ 3 (mod 4)
;
			
  * Aut(𝔽_qℋ_6 )≅ 
			S_8⊕ (SL_2(𝔽_q)^(2)⋊ S_2);
			
  * Aut(𝔽_qℋ_7 )≅
			S_4⊕ (SL_2(𝔽_q)^(3)⋊ S_3),     q≡ 1,7 (mod 8)

			S_4⊕ (SL_2(𝔽_q^2) ⋊ℤ_2) ⊕ SL_2(𝔽_q),    q≡ 3,5 (mod 8)
;
			
  * Aut(𝔽_qℋ_8 )≅
			S_4⊕ (SL_2(𝔽_q)^(3)⋊ S_3),     q≡ 1,3 (mod 8)

			S_4⊕ (SL_2(𝔽_q^2) ⋊ℤ_2) ⊕ SL_2(𝔽_q),    q≡ 5,7 (mod 8)
;
			
  * Aut(𝔽_qℋ_9 )≅
			S_4⊕ (SL_2(𝔽_q)^(3)⋊ S_3),     q≡ 1,7 (mod 8)

			S_4⊕ (SL_2(𝔽_q^2) ⋊ℤ_2) ⊕ SL_2(𝔽_q),    q≡ 3,5 (mod 8)
.
		
		
		
		
		
		
		
		
		
	
	p≠ 2

	
	If p is an odd prime, then, up to isomorphism, the following are non-abelian groups of order p^4 (<cit.>, 117):
	
		
  * 𝒢_1:= ⟨ a, b : a^p^3= b^p=1, ba=a^1+p^2b ⟩;
		
  * 𝒢_2:=⟨ a, b, c: a^p^2 =b^p =c^p=1, cb=a^pbc, ab=ba, ac=ca ⟩;
		
  * 𝒢_3:=⟨ a, b : a^p^2 =b^p^2 =1, ba=a^1+pb  ⟩;
		
  * 𝒢_4:=⟨ a, b, c: a^p^2 =b^p =c^p=1, ca=a^1+pc, ba=ab, cb=bc ⟩;
		
  * 𝒢_5:=⟨ a, b, c: a^p^2 =b^p =c^p=1, ca=abc, ab=ba, bc=cb ⟩;
		
  * 𝒢_6:=⟨ a, b, c: a^p^2 =b^p =c^p=1, ba=a^1+pb, ca=abc, cb=bc ⟩;
		
  * 𝒢_7:=⟨ a, b, c: a^p^2 =b^p=1, c^p=a^p, ab=ba^1+p, ac=cab^-1, cb=bc ⟩,  if  p=3, 
⟨ a, b, c: a^p^2 =b^p =c^p=1, ba=a^1+pb, ca=a^1+pbc, cb=a^pbc ⟩,   if  p>3 ;
		
  * 𝒢_8:=⟨ a, b, c: a^p^2 =b^p=1, c^p=a^-p, ab=ba^1+p, ac=cab^-1, cb=bc ⟩,   if  p=3, 
⟨ a, b, c: a^p^2 =b^p =c^p=1, ba=a^1+pb, ca=a^1+dpbc, cb=a^dpbc ⟩,  if p>3
		
		 d≢0,1 ( mod  p);
		
  * 𝒢_9:=⟨ a, b, c, d: a^p =b^p =c^p=d^p=1, dc=acd, bd=db, ad=da, bc=cb,
           ac=ca,ab=ba ⟩;
		
  * 𝒢_10:=⟨ a, b, c: a^p^2 =b^p=c^p=1, ab=ba, ac=cab, bc=ca^-pb ⟩,  if  p=3,
⟨ a, b, c,d: a^p =b^p =c^p=d^p=1, dc=bcd, db=abd,ad=da,
              bc=cb,ac=ca,ab=ba ⟩,   if  p >3.
		
	
	

	
	For 1 ≤ i ≤ 10,  a complete irredundant set  𝒮(𝒢_i) of strong Shoda pairs of 𝒢_i has been computed in (<cit.>, Theorem 3) which in view of Theorem <ref>, and Eq. (<ref>) yields the following:
	
	
	 The Wedderburn decomposition of 𝔽_q𝒢_i, 1≤ i ≤ 10, is as follows:
		
			
  * 𝔽_q𝒢_1≅𝔽_q⊕𝔽_q^f^((1+p)e)⊕𝔽_q^fp^(pe)⊕ M_p(𝔽_q^fp)^(e);
			
			
  * 𝔽_q𝒢_2≅𝔽_q⊕𝔽_q^f^((1+p+p^2)e)⊕ M_p(𝔽_q^fp)^(e);
			
			
  * 𝔽_q𝒢_3≅𝔽_q⊕𝔽_q^f^((1+p)e)⊕𝔽_q^fp^(ep)⊕ M_p(𝔽_q^f)^(pe);
			
			
			
  * 𝔽_q𝒢_4≅𝔽_q⊕𝔽_q^f^((1+p+p^2)e)⊕ M_p(𝔽_q^f)^(pe);
			
			
  * 𝔽_q𝒢_5≅𝔽_q⊕𝔽_q^f^((1+p)e)⊕𝔽_q^fp^(pe)⊕ M_p(𝔽_q^f)^(pe);
			
			
			
  * 𝔽_q𝒢_6≅𝔽_q⊕𝔽_q^f^((1+p)e)⊕ M_p(𝔽_q^f)^((1+p)e);
			
			
  * 𝔽_q𝒢_7≅𝔽_q⊕𝔽_q^f^((1+p)e)⊕ M_p(𝔽_q^f)^(e)⊕ M_p(𝔽_q^fp)^(e);
			
			
  * 𝔽_q𝒢_8≅𝔽_q⊕𝔽_q^f^((1+p)e)⊕ M_p(𝔽_q^f)^(e)⊕ M_p(𝔽_q^fp)^(e);
			
			
			
  * 𝔽_q𝒢_9≅𝔽_q⊕𝔽_q^f^((1+p+p^2)e)⊕ M_p(𝔽_q^f)^(pe);
			
			
  * 𝔽_q𝒢_10≅{[ 𝔽_q⊕𝔽_q^f^((1+p)e)⊕ M_p(𝔽_q^f)^(e)⊕ M_p(𝔽_q^fp)^(e),                                                  p=3;             𝔽_q⊕𝔽_q^f^((1+p)e)⊕ M_p(𝔽_q^f)^((1+p)e),                                                  p>3 ],
			.
			
		
		
		where o_p(q)=f and e=p-1/f.
		
	
	
	The automorphism group of finite semisimple group algebra of groups of order p^4, p an odd prime, can now be computed similarly.

	
	
	Acknowledgement

	The authors are grateful to I. B. S. Passi and G. K. Bakshi for their valuable comments and suggestions.
	
	
	
	
amsplain
	

 
	
	
