.  

A:

Steinberg's "Lecture Notes on Chevalley Groups" the Corollary of Lemma 54 on page 132. There is possibly a more modern reference.
EDIT: Sorry the reference to Steinberg is not sufficient as he does not treat arbitrary finite reductive groups. However all that is needed is to obtain a slightly more general formula for the order of an arbitrary finite reductive group. This is obtained, for instance, in Corollary 4.2.5 of Geck's book "An Introduction to Algebraic Geometry and Algebraic Groups". the important point is a theorem of Rosenlicht about the fixed points of a connected unipotent group under a Frobenius endomorphism. If $\mathbf{U}$ is a connected unipotent affine algebraic group with Frobenius endomorphism $F' : \mathbf{U} \to \mathbf{U}$ then the finite group $\mathbf{U}^{F'}$ has order $q^{\dim\mathbf{U}}$, (see Geck - Theorem 4.2.4).
Now take $\mathbf{G} = \mathrm{GL}_n(\overline{\mathbb{F}}_p)$ and $F$ to be the map given by $F(x_{ij}) = (x_{ij}^q)$ where $q = p^a$ for some integer $a$ then $\mathbf{G}^F = \mathrm{GL}_n(q)$. The subgroup $\mathbf{B} \leqslant \mathbf{G}$ of upper triangular matrices is an $F$-stable Borel subgroup of $\mathbf{G}$ and its unipotent radical $\mathbf{U} \leqslant \mathbf{B}$ is the subgroup of all upper uni-triangular matrices. This has maximal dimension amongst all connected unipotent subgroups of $\mathbf{G}$, which can be seen in the following way. Assume $\mathbf{V} \leqslant \mathbf{G}$ is a connected unipotent subgroup of $\mathbf{G}$ then as $\mathbf{V}$ is solvable it is contained in a Borel subgroup $\mathbf{B}'$ of $\mathbf{G}$. As $\mathbf{V}$ is a unipotent subgroup of $\mathbf{B}'$ it is contained in the unipotent radical of $\mathbf{B}'$ and as all Borel subgroups of $\mathbf{G}$ are conjugate we have the unipotent radicals of $\mathbf{B}$ and $\mathbf{B}'$ have the same dimension. In particular, we have $\dim
.  

 A: 

 Steinberg's " Lecture Notes on Chevalley Groups " the Corollary of Lemma 54 on page 132. There is possibly a more modern reference. 
 EDIT: Sorry the reference to Steinberg is not sufficient as he does not regale arbitrary finite reductive group. However all that is needed is to obtain a slightly more cosmopolitan formula for the order of an arbitrary finite reductive group. This is obtained, for case, in Corollary 4.2.5 of Geck's book " An Introduction to Algebraic Geometry and Algebraic Groups ". the significant point is a theorem of Rosenlicht about the fixed compass point of a connected unipotent group under a Frobenius endomorphism. If $ \mathbf{U}$ is a machine-accessible unipotent affine algebraic group with Frobenius endomorphism $ farad': \mathbf{U } \to \mathbf{U}$ then the finite group $ \mathbf{U}^{F'}$ has order $ q^{\dim\mathbf{U}}$, (see Geck - Theorem 4.2.4). 
 Now take $ \mathbf{G } = \mathrm{GL}_n(\overline{\mathbb{F}}_p)$ and $ F$ to be the map contribute by $ F(x_{ij }) = (x_{ij}^q)$ where $ q = p^a$ for some integer $ a$ then $ \mathbf{G}^F = \mathrm{GL}_n(q)$. The subgroup $ \mathbf{B } \leqslant \mathbf{G}$ of upper triangular matrices is an $ F$-stable Borel subgroup of $ \mathbf{G}$ and its unipotent radical $ \mathbf{U } \leqslant \mathbf{B}$ is the subgroup of all upper uni - triangular matrix. This has maximal dimension amongst all connected unipotent subgroups of $ \mathbf{G}$, which can be see in the following way. Assume $ \mathbf{V } \leqslant \mathbf{G}$ is a connected unipotent subgroup of $ \mathbf{G}$ then as $ \mathbf{V}$ is solvable it is contained in a Borel subgroup $ \mathbf{B}'$ of $ \mathbf{G}$. As $ \mathbf{V}$ is a unipotent subgroup of $ \mathbf{B}'$ it is hold in the unipotent radical of $ \mathbf{B}'$ and as all Borel subgroups of $ \mathbf{G}$ are conjugate we have the unipotent radicals of $ \mathbf{B}$ and $ \mathbf{B}'$ have the like property. In particular, we have $ \dim
.  

A:

Dteinberg's "Lecture Notes on Chevalley Gtoyps" thx Corolmary of Uemma 54 on page 132. There is posdivly a more modern reference.
DDIT: Sorrj the refwrenre to Steinberg ma not smyficiskt as ie does not trest arbitrasy finite reduwtkvz groups. However all that is needed ys to ontwin a slightly mort gqnerzl formula for the order of an arbjtrary hinite reductivr group. This is obtained, flr ijstance, in Corollagy 4.2.5 of Geck'w bojj "An Introduztion to Algebraic Geojetry and Algebraic Groups". the kmporcant point us a hveorem of Risenlpcht about thc fixed points of a connectec uiiporent group under a Frmbenius endomorphifm. If $\matvby{U}$ is a connected uniporent dffite autebfaid jrohp witj Fcobenius ensomorphism $D' : \mathbf{U} \to \mathbg{U}$ nnen the finife grotp $\mathbf{U}^{F'}$ has order $q^{\dim\mathbf{U}}$, (see Gecn - Fheorem 4.2.4).
Now take $\mathbf{T} = \mathrm{GL}_n(\overline{\mwthbb{F}}_p)$ agd $F$ to be the map given by $F(x_{ij}) = (x_{ij}^q)$ where $q = p^d$ for romt lktegde $w$ then $\mathbf{G}^F = \mathrm{GL}_n(q)$. The subgroup $\mathfr{B} \lvqslant \mathbf{G}$ on upper triangular mwttyces is an $F$-sjable Borem subgroup of $\mathhf{G}$ and its ynipotent radocal $\mathbf{U} \leqslant \mathbd{B}$ is the suygriup of all upper uui-triangular mattices. Yhis has maximal dimensnon amkngst all clnnected hvipotent subgrouos pf $\mathbf{G}$, which can be seen in the fillocing way. Assome $\matrbf{V} \leqslwnt \mathbf{G}$ is a connectfd unnpotett subgrouo of $\mathbf{G}$ then as $\mathbf{V}$ is solvable it ix wonnained in a Bovel subgroup $\maehbf{B}'$ of $\mathby{G}$. As $\machbf{V}$ ks a uniponent subgcoup of $\mathff{B}'$ it is condwined in the unipotegt rqdicql of $\mxghbf{B}'$ and as akl Borel subgroups if $\mathbf{G}$ are conmugatg se have the uni'uuebt radicals of $\maghbs{B}$ aid $\maevbf{B}'$ have tha saoe aomenskon. In kcrtlcuuar, ee have $\dim
. A: Steinberg's "Lecture Notes on Chevalley Corollary Lemma 54 page 132. There reference. Sorry the reference Steinberg is not as he does not treat arbitrary reductive groups. However all that is needed is to obtain a slightly more formula for the order of an arbitrary finite reductive group. This is obtained, instance, Corollary of book "An Introduction to Algebraic Geometry and Algebraic Groups". the important point is a theorem of about the fixed points of a connected unipotent under a Frobenius endomorphism. $\mathbf{U}$ is a connected unipotent algebraic with Frobenius $F' \mathbf{U} \mathbf{U}$ then the group $\mathbf{U}^{F'}$ has order $q^{\dim\mathbf{U}}$, (see Geck - Theorem 4.2.4). Now take $\mathbf{G} = \mathrm{GL}_n(\overline{\mathbb{F}}_p)$ and $F$ be the by $F(x_{ij}) (x_{ij}^q)$ $q p^a$ for some then $\mathbf{G}^F = \mathrm{GL}_n(q)$. The subgroup of upper triangular matrices is an $F$-stable Borel of $\mathbf{G}$ its unipotent radical $\mathbf{U} \leqslant \mathbf{B}$ the subgroup of all upper uni-triangular matrices. This maximal dimension amongst all connected unipotent subgroups of $\mathbf{G}$, which can be seen in the Assume $\mathbf{V} \leqslant \mathbf{G}$ a connected unipotent of then $\mathbf{V}$ solvable it contained in a Borel subgroup $\mathbf{B}'$ of $\mathbf{G}$. As $\mathbf{V}$ is unipotent subgroup of $\mathbf{B}'$ it is contained in the unipotent $\mathbf{B}'$ as all Borel of $\mathbf{G}$ are conjugate have unipotent radicals of $\mathbf{B}$ have same we $\dim
.  

A:

Steinberg's "Lecture Notes on chevalley GRoups" The corOlLary Of LeMma 54 on page 132. There IS posSibly a more modern refereNce.
EDiT: sOrry THe ReferEnce to STEiNBErg Is NoT suFfICiEnt as He dOes not tReat arbitrAry FiNite reductivE GrOups. HoweveR alL that is needeD is To obtaIn A slIGhtly MorE geneRal forMUla for The order oF aN ArbitrARy finitE REdUctiVe group. This is obtaINeD, For instance, in COrollaRy 4.2.5 OF GECK's bOok "an IntroducTiOn to ALGebraic gEoMETRy aND Algebraic GroUps". the imporTAnt Point iS a TheORem of ROsenlIcHT abOut the fixed PoinTs of a connEcted uNIpotent GRoup undEr a FroBenIus EndoMOrPhIsm. if $\MAthBF{U}$ Is a COnnEcted uniPoTeNt affIne aLGEBRaic GroUp wiTh FroBenius endomorPhiSm $F' : \mAThbF{U} \to \mAthbf{u}$ theN tHe finIte groUp $\matHbF{U}^{F'}$ has order $q^{\dim\MathBf{U}}$, (see GecK - ThEoRem 4.2.4).
noW take $\MAthbf{G} = \MatHrm{gL}_n(\overLine{\matHBb{F}}_P)$ aND $f$ To Be the map given by $F(x_{iJ}) = (x_{IJ}^Q)$ wHere $q = p^a$ fOr some INtEgER $a$ then $\maThBf{G}^f = \matHRM{GL}_n(q)$. the sUBgRoup $\mathBf{B} \leqSLaNt \Mathbf{G}$ Of Upper tRiAngUlaR matrICes iS an $F$-stAble BoreL subgROup of $\mathbf{G}$ anD Its unipotent rADiCAL $\mAThbf{u} \leQslant \mathbF{B}$ is THe suBgroUP oF alL Upper Uni-trIaNGuLAr matrices. This has maXiMal dimEnsioN amongst all coNnected uniPOTEnt subgrOups OF $\mAThbf{G}$, which can bE seen In the folloWIng way. AsSume $\mAthbf{V} \leQslant \matHBF{G}$ is a conNecTed UniPotENT sUbgroup of $\mathBF{g}$ theN aS $\mathbf{v}$ is SolvablE it Is cOntAinEd In a Borel sUbgroup $\mAtHbF{B}'$ Of $\MatHbf{G}$. AS $\Mathbf{V}$ iS a UniPoTenT subgROup of $\mAthbf{b}'$ it iS cOnTAinEd in the UNiPOTent RaDiCal oF $\maThBf{B}'$ anD as aLL BoRel subgRoups of $\maThbF{g}$ are CoNjUgate we Have the unipotEnT radicals oF $\mAthBf{B}$ and $\MAThbf{B}'$ havE the same dimension. In partICular, we HavE $\dim
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A:

Steinberg's "Lect ure Noteson Ch eva lle yGrou ps"the Corollaryo f Le mma 54 on page 132. Th ere i sp ossi b ly  a mo re mode r nr e fer en ce .
E DI T :Sorry  th e refer ence to St ein be rg is not su f fi cient as h e d oes not trea t a rbitra ry  fi n ite r edu ctive  group s . Howe ver all t ha t  is ne e ded ist o  o btai n a slightly more ge n eral formula f or the  o r de r ofanarbitraryfi niter eductiv e  g r o u p.T his is obtain ed, for ins t anc e, inCo rol l ary 4. 2.5 o fG eck 's book "An  Int roduction  to Al g ebraicG eometry  and A lge bra ic G r ou ps ".th e  im p or tan t  po int is a  t he oremof R o s e n lich t a bout  thefixed pointsofa co n nec ted u nipot entgr oup u nder a  Frob en ius endomorphis m. I f $\mathb f{U }$  is  a  conn e cted u nip ote nt affi ne alge b rai cg r o up  with Frobenius en do m o rp hism $F'  : \ma t hb f{ U } \to \m at hbf {U}$ t hen t he f i ni te group  $\mat h bf {U }^{F'}$  h as ord er  $q ^{\ dim\m a thbf {U}}$,  (see Ge ck -T heorem 4.2.4). Now take $\ma t hb f { G} = \m ath rm{GL}_n(\o verl i ne{\ math b b{ F}} _ p)$ a nd $F $t ob e the map given by$F (x_{ij }) =(x_{ij}^q)$ w here $q =p ^ a $ for so me i n te g er $a$ then $\ mathb f{G}^F = \ m athrm{GL }_n(q )$. Thesubgroup$ \ mathbf{B } \ leq sla nt\ m at hbf{G}$ of up p e r tr ia ngularmat rices i s a n $ F$- sta bl e Borel s ubgroupof  $ \m at hbf {G}$a nd its u ni pot en t r adica l  $\mat hbf{U } \l eq sl a nt\mathbf { B} $ is t he  s ubgr oup  o f all  upp e r u ni-tria ngular ma tri c es.Th is  has ma ximal dimensi on  amongst a ll  co nnecte d unipoten t subgroups of $\mathbf { G}$, wh ich  canbe s een in th e f ollowi ngw ay. As sume $ \math bf {V} \ leqsl a n t\ma th bf{G}$ isa con necte dunip otent s ubgroup of $\mathb f {G} $ then as $\m ath bf{V } $  i s s o lv a ble  i t  is c ontained in a B orel subgr ou p  $ \mathbf{B} ' $ o f$\mathb f{G}$.As $\ m athbf{V }$ is a u nipotentsu bgro u p  of  $\mathbf{ B}'$ itis contai n ed in th e uni pot ent ra di cal  of $ \mathb f {B} '$ an d as a ll  Borel  subg ro ups of $ \mathbf{G}$ are conjuga te wehavethe  unipoten t r a dic als of $\ math bf{B}$ and  $\ mat hbf{B }'$ havethes am e d i mensi on.I n particu l ar , w e ha ve $\dim
. _

A:

Steinberg's "Lecture_Notes on Chevalley Groups"_the Corollary_of_Lemma 54_on_page 132. There_is possibly a_more modern reference.
EDIT: Sorry_the reference to_Steinberg_is not sufficient as he does not treat arbitrary finite reductive groups. However all_that_is needed_is_to_obtain a slightly more general_formula for the order of_an arbitrary_finite reductive group. This is obtained, for instance,_in_Corollary 4.2.5 of_Geck's book "An Introduction to Algebraic Geometry and Algebraic_Groups". the important point is a_theorem of Rosenlicht_about_the_fixed points of a_connected unipotent group under a Frobenius_endomorphism. If $\mathbf{U}$ is a connected_unipotent affine algebraic group with Frobenius endomorphism_$F' : \mathbf{U} \to \mathbf{U}$ then_the finite group $\mathbf{U}^{F'}$ has_order $q^{\dim\mathbf{U}}$,_(see Geck - Theorem 4.2.4).
Now_take $\mathbf{G} =_\mathrm{GL}_n(\overline{\mathbb{F}}_p)$ and_$F$ to be_the map given by $F(x_{ij}) =_(x_{ij}^q)$ where $q_= p^a$ for some integer $a$_then_$\mathbf{G}^F = \mathrm{GL}_n(q)$._The_subgroup_$\mathbf{B} \leqslant_\mathbf{G}$ of upper_triangular_matrices is_an_$F$-stable Borel subgroup of $\mathbf{G}$ and_its_unipotent radical $\mathbf{U} \leqslant \mathbf{B}$ is the_subgroup of all upper_uni-triangular_matrices. This has maximal_dimension amongst all connected unipotent_subgroups of $\mathbf{G}$, which can be_seen in_the following_way. Assume $\mathbf{V} \leqslant \mathbf{G}$ is a connected unipotent subgroup of_$\mathbf{G}$ then as $\mathbf{V}$ is solvable_it is contained in_a Borel_subgroup_$\mathbf{B}'$ of $\mathbf{G}$._As_$\mathbf{V}$ is_a unipotent subgroup of $\mathbf{B}'$ it is_contained in_the unipotent radical of $\mathbf{B}'$ and_as all Borel subgroups_of_$\mathbf{G}$ are conjugate we have the_unipotent radicals of $\mathbf{B}$ and $\mathbf{B}'$_have the same dimension. In_particular,_we_have $\dim