Representations and Characters of Groups (2nd Edition) by Martin Liebeck, Gordon James

By Martin Liebeck, Gordon James

This is the second one variation of the preferred textbook on illustration conception of finite teams. The authors have revised the textual content enormously and integrated new chapters on Characters of GL(2,q) and variations and Characters. the speculation is constructed by way of modules, considering that this can be acceptable for extra complicated paintings, yet massive emphasis is positioned upon developing characters. the nature tables of many teams are given, together with all teams of order below 32, and all yet one of many easy teams of order below one thousand. every one bankruptcy is followed via various workouts, and entire strategies to all of the routines are supplied on the finish of the book.

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9 Example Let G ˆ S4 and let B denote the basis v1 , v2 , v3 , v4 of V. 10 De®nition Let G be a subgroup of Sn . The FG-module V with basis v1 , . . , v n such that v i g ˆ v ig for all i, and all g P G, is called the permutation module for G over F. We call v1 , . . , v n the natural basis of V. Note that if we write B for the basis v1 , . . , v n of the permutation module, then for all g in G, the matrix [ g]B has precisely one nonzero entry in each row and column, and this entry is 1. Such a matrix is called a permutation matrix.

First, the fact that r is a homomorphism shows that v(( gh)r) ˆ v( gr)(hr) for all v P V and all g, h P G. Next, since 1r is the identity matrix, we have v(1r) ˆ v for all v P V. Finally, the properties of matrix multiplication give (ëv)( gr) ˆ ë(v( gr)), 38 FG-modules 39 (u ‡ v)( gr) ˆ u( gr) ‡ v( gr) for all u, v P V, ë P F and g P G. 2(1). Thus     1 0 0 1 X , br ˆ ar ˆ 0 À1 À1 0 If v ˆ (ë1 , ë2 ) P F 2 then, for example, v(ar) ˆ (Àë2 , ë1 ), v(br) ˆ (ë1 , Àë2 ), v(a3 r) ˆ (ë2 , Àë1 )X Motivated by the above observations on the product v( gr), we now de®ne an FG-module.

Ws is a basis of W. Show that V ˆ U È W if and only if u1 , . . , ur , w1, . . , ws is a basis of V. 5. (a) Let U1, U2 and U3 be subspaces of a vector space V, with V ˆ U1 ‡ U2 ‡ U3. Show that V ˆ U1 È U2 È U3 D U 1 ’ (U 2 ‡ U3 ) ˆ U 2 ’ (U 1 ‡ U3 ) ˆ U 3 ’ (U 1 ‡ U2 ) ˆ f0gX (b) Give an example of a vector space V with three subspaces U1, U2 and U3 such that V ˆ U1 ‡ U2 ‡ U3 and U1 ’ U 2 ˆ U1 ’ U3 ˆ U2 ’ U 3 ˆ f0g, but V Tˆ U1 È U2 È U3. 6. Suppose that U1, . . , Ur are subspaces of the vector space V, and that V ˆ U1 È .

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