M.A. MATHEMATICS GROUP PROPERI'IES WITH THE AID OF REPRESENTATION THEORY A Thesis by Pushp Lata Jain Submitted to the Faculty of Graduate Studies and Research in partial ful­ fil.Iœnt of the requirements for the degree of Master of Arts McGill University, Montreal, Canada. July, 1960 ACKNOWLEDGEMENTS The author expresses her sincere gratitude to Dr. M.D. Burrow for his guidance and encouragement during the course of this work. No other assistance was received. TABLE OF CONTENTS 1. Introduction 2. Representation Modules m • • • • • • • • • • • • • • • • • • • • 3 Realization of the Representation of a Ring into Linear Transformation of '1'7i. as Matrices • • • • • • • • • • • • • • • 4 Reducibility and Decomposability of 17!. • • • • • • • • • • • • • 5 3. The Regular Representation or a Special Representation Module and Its Properties • • • • • • • • • • • • • • • • • • • • 8 Form of the Regular Representation of a Semi-simple Ring • • • • • 21 Multiplicity of Principal Indecomposable Components in the Regular Representation • • • • • • • • • • • • • • • • • • • • 25 4. Applications to Groups • • • • • • • • • • • • • • • • • • • • • • 29 Group Algebra 1 and a Close Connection between the Representations of r and the Group • • • • • • • • • • • • • • 29 The Trace Function ) on r 1 Simplification of this Function when Restricted ~o the Group • • • • • • • • • • • • • • • • • 30 The Number of Distinct Irreducible Representations of r and G (group) ••••••••••••••••••••• • • • • 33 Character Relations • • • • • • • • • • • • • • • • • • • • • • • 36 The Properties of Groups by an Application of the Theory • • • • • • • • • • • • • • • • • • • • • • • • • • • • 42 5. Concluding Re marks • . • • • • . .• • • • • • . • • • 57 6. Bibliography • • • • • • • • • • • • • • • • • • • • • • • • • 59 GROUP PROPERI'IES WITH THE AID OF REPRESENTATION THEORY Section l Introduction A representation of degree n of a group is a hororrorphisrn of the group into a group of nxn matrices 1dth coefficients in a given field K. The image group of matrices is necessarily a sub-group of GL (K), the group of all non-singular ~~ matrices with coefficients in n K. Two representations ~ and ~, necessarily of the same degree n, are equivalent if there is a matrix T € GLn (K) such that T 1(g)T;"~g) 1for every g f G. Equivalent representations are to be regarded as essentially the sarne. The representation ~ is said to be reducible if ~ is equi- valent to ~ and the matrix ~(g) is of the form: A(g) 0 yg c;G , ~(g) = * B(g) where A(g) is an IlDCIJl matrix, B(g) is an (n - rn) x (n - m) matrix while 0 stands for the ~-m) zero matrix and ~~ is sorne (n - m) x rn ma.trix. If ~ is reducible in this way it is clear that ~l and ~2 defined thus: ~l (g) = A(g), ~2 (g) = B(g) provide representations of G. Thus a reducible representation gives rise to representations of smaller degree. A representation which is not reducible is called irreducible. An important objective of the theory of representations is the survey of all inequivalent irreducible representations of a given group. By considering the traces of the matrices of a representation €: , a numerical function fon the group is defined: f(g) =trace ~(g). Since trace (T ~(g)T-1 ) = trace ~(g) it is apparent that equivalent representations give rise to the same nurnerical functions on the group. Of special importance are the function J, the so-called character--3, derived from the irreducible representations: ~(g) = trace ~(g), ~ irreducible. As we will see, important properties of a group can often be deduced from a knowledge of its characters. Indeed there are results in group theory which have never been proved in any other way. On the other hand, the characters can be determined when relatively little is known about the group so that the representation theory is an invaluable aid to group theory. The classi cal theory, the case where the given field K is the 1 field of complex numbers, is due to G. Frobenius • His most important 2 results were proved independently by Burnside who gave a number of important applications as well as its classical treatrnent. Fundamental work in this field was also done by I. S chu~. An important treatment of the theory can be given which is closely related to the representation theory of hypercornplex numbers4 considered by E. Noether. 5" In this t hesis we atternpt to give an account of the theory of representations from a consistently algebraic point of view. \~ile the results are weil known, many of the proofs involved in t his approach appear only in the journals or in lecture notes, notably of R. Brauer, 3 but so far as the writer is aware are not to be found in any published textbook. Section 2 Representation Hodules Let 1J be a ring and K be a field. By a representation module rrc.. we mean a (K, '0' )-rodule )'i( , that is an additive abelian group nt.. having the elements of K and 1J as left and right operators respectively. This means that 'd u (;- rrt., V a ~ K, V at "()" there are unique eleJœnts au ~nt and ua (;;rn and the following rules hold: 1) a(u + v) au + av 2) (a + 13)u ;: au + !3u 3) (a.j3)u ;: a(13u) 4) (u + v)a = ua+ va 5) u(a + b) ;: ua+ ub 6) u(ab) = (ua)b 7) (au)a = a.( ua) From now on we will asswne that '0"' is a finite algebra over K with the identity 1 and that furthermore 8) a(ua) = u(aa) u(aa). For every a t '0' we can define a mapping ~(a): m -~ rn thus: 'V u é-llt , u ~ (a) = ua. · · (u + v) ~ (a ) = (u + v)a = ua + va = u ~ (a) + v~ (a) and (ku) ~ (a) = (ku) a = k( ua) = k( u 0::: (a)). 4 We see that ~(a) is a linear transformation in -,re. regarded as a K-vector space. Moreover since u ~(a+b) .. u(a + b) = ua + ub = u~(a) + u~(b) =~(a)+ ~(b)) and u ~(ab) = u(ab) = (ua)b ~ (ua) c (b) = u(_Ma) ~(b)) We have ~(a + b) =~(a) +~ (b) and a:;: (ab) = ~(a) ~ (b). Hence the mapping a-P da) is a ring-homomorphism. In this way each representation module ~ives rise in a natural way to a hommorphism !:'of the ring ~into the ring of linear transformations in -rrt regarded as a vector space. Now if ~has a finite K-basis (u , u , ••• un), then there 1 2 are uniquely determined elements aiJ ~ K such that n (i = l, 2, ••• , n) and we arrive at a realization of c- (a) as an (nxn) matrix (a J). It 1 is easi.ly sean that the correspondence ~(a) - /A = (aiJ) is an isom:>rph- ism of the ring !::'"'("'0"') onto a ring of nm matrices. Moreover if v , ••• , v 1 0 is another basis of ~connected with the first basis by the non-singular -1 ..... ) matrix T = (t1J); T ~ (tiJ ; n v. = ~ t.JUJ (i = 1, 2, ••• , n) ~ J==l ~ n n then since vi ~(a) = via= z tiJuJa '"' ~ tirjKu.k J=:l J,k==l 5 n ~ J,k,S=l the new correspondance is ~(a) -~ TAT-l To sum up: each representation module n1. provides a represent- ation ~ of 1)' into linear transformation of nt which can in turn be realized as matrices determined uniquely up-to equivalence. More generally it is easily seen that operator-isomorphic(a) representation modules lead to equivalent representations. A representation module )')"t is reducible if n <! m and rt.. is a proper (K, -&) submodule of m. In this case 1"i.. itself and the factor module ~ are representation modules. The corresponding representation~ re. provided by f'l'C is said to be reducible and the representations a'J o_::- , 2 provided by ~ and ~ respectively are called top and bottom constituants "'H If mis reducible and we take a basis of '17t adapted to n, i.e. a ba sis \ ul' u , ••• , of 'h"t such that u , u , ••• , uT'J is a basis 2 un~ ~ 1 2 of )lt then the matrix corresponding to o_:::(a) is of the form: A(a) 0 ) o:_(a) = { * B(a) where A(a) is the rxr matrix corresponding to~(a) and B(a) is the Ln-~-~ matrix corresponding to ~2 (a). These results lead us back to the similar remarks of Section l. 6 )1(. is decomp:>sable if Tit = 1'r! + nt • 1 2 In this case the corresp:>nding representation 0::: of mis said to be decomposable into the representations o_: and o.::, called comp:ments, 1 2 corresp:>nding to the representation .m:>dules and m respectively. 11'\ 2 \'lith a suitable basis the corresponding matri.xa:::(a) has the form 0:::. (a) = ( A(a) 0 ) 0 B(a) It is to be noted that ~may be indecomposable, that is not decomposableJ without being irreducible. Since l'Y't is a group with operators, we have from group theory the following results. Il Theorem 2-1: (Jordan Holder) For any composition series of subrodules of 111. ; (l) m. the number k and the factor modules --~- , i = 1, 2, ••• , k, are uniquely rrt. 1 ~- determined, the latter up to operator-isomorphism. Theorem 2-2: (Remak-Krull-Schmidt) For any decomposition of ~ into indecomposable submodules: . (2) + m,r the number r and the indecomposable submodules nti are uniquely deter­ mined, the l at ter up to oper ator-isomorphi sm. 7 Theorem 2-3: (Fittings Lemma,or rather a special case of it) If '"'is indecomposable and both chain conditions hold, then any operator endo- m:>rphism of nt is either nilpotent or an isomorphism. In terms of representation theory these results give: I' _ Every representation ~ of a ring -& has a unique number of unique irreducible constituents o:J; i = 1, 2, ••• , k. Also if a suitable basis of ~ is chosen such that it is adapted to the chain of submodules in (1) ~(a) will be of the form At a) ~(a): * Vfuere A.