By Wehrfritz B.A.F.

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M). Put u, = v — v,„ (j = 1, 2, ... , m —1). ,u,,, -1] is an (m —1)-dimensional G-module. In particular, when m = 4, compute the 3 x 3 matrices that describe the action of T = (12), p = (123), A = (12)(34), y = (1234) on U and obtain the character value in each case. 6. Suppose the group G has a matrix representation A (x) of degree two over the rationals, with the property that, for a certain central element z of G A (z )=(^ ^). Prove that A is reducible over the rationals. 7. Let V be the m-dimensional vector space over C, consisting of all row-vectors with m components.

32) becomes z = Aix; = E A u 'x u. 33) Thus conjugate basis elements have the same coefficient in z. Suppose the k conjugacy classes of G are listed as follows: -1 C1 = 111, C2 = {y2, P2- Y2P2 , g2 Y2 g 2, .. }, Ck = tYk, P k l YkPk, qk l Ykgk, • • •}• We associate with Ca the element Ca =Ya +P: l yapa +g -l ya ga +... 34) of Gc. 33) as z —y 1 C 1 +y 2 c 2 + • • . 35) where each y is one of the As. Conversely, we have that u ' c u = c (a =1, 2, ... , k}, - Œ Œ because the element on the left consists of the same terms as Ca , though possibly in a different order.

Suppose the image of 1 under 0 is the element t of Ge , that is 10 = t. 27) and leaving x arbitrary we find that tx =x0 (x E G). 20) we find that vO = tv. 27) holds. 28) sets up a one-to-one correspondence between the elements of ge and those of G c : every endomorphism of the G-module G c is equivalent to the left-multiplication by a fixed element of G c . 28) is an isomorphism between the vector spaces and G c . For, if analogously where n E s E G c , then v(a0+b77)= (at +bs)v (a, b E C). It follows that dim A = dim Gc = g.