By Mary W Gray
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7. Prove that (H x 'K) xL=:: H x K x L=:: H x (K xL). 8. An abelian group of exponent p is a direct product of cyclic groups of order p-such groups are called elementary abelian p-groups. ] *9. (The mapping property of the cartesian product). Let G = Cr, eA G, and define the projections 7t,: G -+ G, by setting x n , equal to the A-component of x . Show that 7t, is a homomorphism. Let there be given a family of homomorphisms qJ,: H -+ G, from some group H . Prove that there exists a unique homomorphism qJ: H -+ G such that qJ7t, = qJ, for all A.
Write IGI = pam where the integer m is not divisible by p. (i) Every p-subgroup of G is contained in a subgroup of order pa. In particular, since 1 is a p-subgroup, Sylow p-subgroups always exist. (ii) If np is the number of Sylow p-subgroups, np == 1 mod p. (iii) All the Sylow p-subgroups are conjugate in G. Proof. / be the set of all subsets of G with exactly pa elements. / by right multiplication, so we have a permutation repre- 40 1. Fundamental Concepts of Group Theory sentation of G on :/ with degree Let us show that p does not divide n.
Proof. Let x be any element of X . 2. But StG(x)
A radical approach to algebra by Mary W Gray