Group Theory

Download e-book for iPad: A first course in noncommutative ring theory by T.Y. Lam

By T.Y. Lam

ISBN-10: 0387975233

ISBN-13: 9780387975238

ISBN-10: 3540975233

ISBN-13: 9783540975236

By means of aiming the extent of writing on the beginner instead of the gourmand and through stressing the function of examples and motivation, the writer has produced a textual content that's compatible for a one-semester graduate direction or for self-study.

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Identify v1 E Vi with (v1,0) EV and v2 E Vi with (O,v2) E V. Then Vi and V2 are embedded in V, so we can consider V as the inner direct sum Vi E9 Vi of its submodules Vi, Vi. If Ti E End(l/i), i = 1, 2, then the operator representation of G in V is called the direct sum of T1 and T2 and denoted by T1 + T2. Similarly, we can define the sum of any finite number of representations. Let B1 = {vi}i and B2 = {vi}~+l be bases of Vi and Vi, respectively, Ti the matrix representation of Gin l/i, written in terms of the basis Bi (i = 1, 2).

Therefore a q'-Hall subgroup F of G centralizes Qi. Since Q is abelian, this D implies Z(G) n Q =Qi and G = (Q n Z(G)) x FQ2. LEMMA 19. Suppose that G = PQ,E(pn) Sylq(G), Q <1 G. If Z(G) = {1}, then n:::; T· ~ P E Sylp(G), E(qm) ~ Q E Proof. By Maschke's Theorem, Q = Ri x · · · x Rs, where Ri, ... , Rs are minimal normal subgroups of G. Denoting Pi = Z(PR;), we see that Pi is a p-subgroup, and IP: Pil = p for each i by Schur's Lemma (recall that Ri is a minimal normal subgroup of PRi)· Since Pis an elementary abelian group, it follows that P = Pi x Ci, where Ci is a subgroup of order p, i = 1, ...

Z(Fn) = {aln la E F}, where In is the n x n identity matrix. Deduce that Z(GL(n, F)) = {aln I a E F*}. We might have considered the analogs of Lemmas 4 and 5 for groups. But the context of algebras is more general, due to the existence of group algebras. In particular, we have the following result. THEOREM 6. The center of an irreducible finite group of matrices over F is cyclic. If F is algebraically closed, this center consists of scalar matrices. Indeed, Z(G), being an abelian subgroup of the multiplicative group of the skew field L = CFJG), is cyclic.

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A first course in noncommutative ring theory by T.Y. Lam

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