2-Generator conditions in linear groups by Wehrfritz B.A.F.

By Wehrfritz B.A.F.

Show description

Read or Download 2-Generator conditions in linear groups PDF

Best symmetry and group books

Travail, affection et pouvoir dans les groupes restreints (French Edition)

Les groupes restreints ont été et seront au coeur de nos vies pendant toute notre life et, pourtant, c'est depuis moins d'un siècle que chercheurs et théoriciens se penchent sur les rouages complexes de leur développement. Cet ouvrage suggest un modèle qui feel los angeles présence, dans tout groupe restreint, de trois zones dynamiques, les zones du travail, de l'affection et du pouvoir.

Additional info for 2-Generator conditions in linear groups

Sample text

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.

Download PDF sample

Rated 4.71 of 5 – based on 24 votes