By Ya. G. Berkovich and E. M. Zhmud

This e-book discusses personality thought and its functions to finite teams. The paintings locations the topic in the achieve of individuals with a comparatively modest mathematical heritage. the mandatory historical past exceeds the traditional algebra direction with appreciate merely to finite teams. beginning with simple notions and theorems in personality thought, the authors current quite a few effects at the homes of complex-valued characters and functions to finite teams. the most subject matters are levels and kernels of irreducible characters, the category quantity and the variety of nonlinear irreducible characters, values of irreducible characters, characterizations and generalizations of Frobenius teams, and generalizations and purposes of monomial teams. The presentation is distinct, and plenty of proofs of recognized effects are new. many of the leads to the booklet are provided in monograph shape for the 1st time. various routines provide more information at the subject matters and support readers to appreciate the most thoughts and effects.

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EXERCISE 18. Let T be a faithful completely reducible representation of a group G. If all the irreducible components of T are one-dimensional, G is abelian. Indeed, let Ti, ... , Tk be all the irreducible components of T. Then {1} = kerT = n~=l kerTi, whence G is isomorphic to a subgroup of the direct product of the cyclic groups G/ kerTi, i = 1, ... , k (Theorem 6). EXAMPLE. Let B be a nonzero subgroup of the additive group p+ of a field F, G= { (~ ~)lbEB}. Then we can regard G as a group of linear operators of the space F 2 .

F-space V, and B' = {vaf bases of the n vj = I:>ijVi (j E {l, ... ,n}), i=l so that S = (a ij) is the matrix of the transformation from B to B'. , T'. Since we can view any element of GL(n, F) as a transformation matrix, T yields the entire equivalence class of matrix representations of G. Hence, any G-module determines a class of matrix representations. EXERCISE 12. Isomorphic G-modules determine equivalent matrix representations and, consequently, equivalent operator representations determine equivalent matrix representations.

A+ b = L(a(x) + b(x))x xEG (from now on we write (A3) Lx instead of LxEG); = 'Ya(x) (ab)(g) = L (l'a)(x) (A4) ('YE F, x E G), a(x)b(y), xy=g that is, ab= L a(x)b(y)g = La(x)b(y)xy xy=g x,y (collecting similar terms in the last sum). Suppose that V is a G-module and put av= La(x)(xv) (a EFG, v EV). x This enables us to view V as an FG-module. Identifying an element g of G with the element Lx Dx,gX of FG, we may consider Gas a subset of FG. It is clear that §5. MATRIX REPRESENTATIONS 7 any FG-module can be considered as a G-module and, conversely, any G-module can be considered as an FG-module.

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