In mathematics, Clifford theory, introduced by , describes the relation between representations of a group and those of a normal subgroup. Alfred H. Clifford Alfred H. Clifford proved the following result on the restriction of finite-dimensional irreducible representations from a group G to a normal subgroup N of finite index: Clifford's theorem Theorem. Let π: G → GL(n,K) be an irreducible representation with K a field. Then the restriction of π to N breaks up into a direct sum of irreducible representations of N of equal dimensions. These irreducible representations of N lie in one orbit for the action of G by conjugation on the equivalence classes of irreducible representations of N. In particular the number of pairwise nonisomorphic summands is no greater than the index of N in G. Clifford's theorem yields information about the restriction of a complex irreducible character of a finite group G to a normal subgroup N. If μ is a complex character of N, then for a fixed element g of G, another character, μ(g), of N may be constructed by setting for all n in N. The character μ(g) is irreducible if and only if μ is. Clifford's theorem states that if χ is a complex irreducible character of G, and μ is an irreducible character of N with then where e and t are positive integers, and each gi is an element of G. The integers e and t both divide the index [G:N]. The integer t is the index of a subgroup of G, containing N, known as the inertial subgroup of μ. This is and is often denoted by The elements gi may be taken to be representatives of all the right cosets of the subgroup IG(μ) in G. In fact, the integer e divides the index though the proof of this fact requires some use of Schur's theory of projective representations. Proof of Clifford's theorem The proof of Clifford's theorem is best explained in terms of modules (and the module-theoretic version works for irreducible modular representations). Let K be a field, V be an irreducible K[G]-module, VN be its restriction to N and U be an irreducible K[N]-submodule of VN. For each g in G, U.g is an irreducible K[N]-submodule of VN, and is an K[G]-submodule of V, so must be all of V by irreducibility. Now VN is expressed as a sum of irreducible submodules, and this expression