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C 1, we have fl = 0 and hence a' = 0 for some c. Therefore 0 = ct = x(g)/x(l) and x(g) = 0. The proof is complete. 9) THEOREM Let G be a nonabelian simple group. Then { 1) is the only conjugacy class of G which has prime power size. Proof Suppose g E G , I Cl(g)I = p", and g # 1. Let x E Irr(G), x # lG. Then ker x = 1 since G is simple and Z(x) = Z(G) = 1 since G is nonabelian. 8. Now 0 = P(g) = c x(l)x(g) = 1 + ,x~lrr(G) c x(l)x(g). ,xeIrr(G): plx(1) We have - 1 = pa, where the sum being taken over x E Irr(G) where p I x( 1).

Show that Q n Z(x) # 1 and consider det x. 8) Let x be a (possibly reducible) character of G which is constant on G - { l}. Show that x = a l G bp,, where a, b E Z and p is the regular character of G . Show that if G # ker x, then x( 1) 2 I G I - 1. + Problems 45 Hint First show that x = al, + bp, for some a, b E C. 9) Let XI, . , x k be the conjugacy classes of a group G and let K . . , Kk be the corresponding class sums. Choose representatives g iE X i and let aijvbe the integers defined by ,, K i K j = C aijvK,.

The fact that x + $ is a character is a triviality. We may define a new class function x$ on G by setting (x$)@) = x(g)$(g). It is true but somewhat less trivial that x$ is a character. We shall construct a new C[G]-module V 0 W called the tensor product of V and W. Choose bases {ulr . . , u,} for V and {wl,. ,w,} for W . Let V 0 W be the @-spacespanned by the mn symbols ui @I w j . Wedefine u 0 w = aib,{ui 0 wj)E V 0 W. 1 1 c Note that not every element of V @I W has the form u 0 w for u E I/ and w E W (except in the special case that n or rn = 1).