# Download An Introduction to the Theory of Groups by Joseph J. Rotman PDF

By Joseph J. Rotman

Fourth Edition

J.J. Rotman

An creation to the idea of Groups

"Rotman has given us a truly readable and helpful textual content, and has proven us many attractive vistas alongside his selected route."—MATHEMATICAL REVIEWS

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Extra resources for An Introduction to the Theory of Groups

Example text

Proof. Define a function cp: S x T --+ ST by (s, t) H st. Since cp is a surjection, it suffices to show that if x E ST, then Icp-1(x)1 = IS n We show that cp-1(X) = {(sd,d- 1t): dES n T}. It is clear that cp-1(X) contains the right side. For the reverse inclusion, let (s, t), (a, f) E cp-1(X); that is, s, a E S, t, f E T, and st = x = af. Thus, s-la = tf- 1 E S n T; let d = s-la = tf- 1 denote their common value. Then a = S(S-l a) = sd and d- 1t = felt = f, as desired. • n There is one kind of subgroup that is especially interesting because it is intimately related to homomorphisms.

A right coset St has many representatives; every element of the form st for s E S is a representative of St. The next lemma gives a criterion for Lagrange's Theorem 25 determining whether two right co sets of S are the same when a representative of each is known. 8. If S :-;::; G, then Sa ifb- 1 aES). = Sb if and only if ab- 1 E S (as = bS if and only Proof. If Sa = Sb, then a = 1a E Sa = Sb, and so there is s E S with a = sb', -1 hence, ab = s E S. Conversely, assume that ab- 1 = rr E S; hence, a = rrb.

58. Let M be a maximal subgroup of G; that is, there is no subgroup S with M < S < G. 59 (Schur). Let f: G ..... H be a homomorphism that does not send every element of G into 1. If G is simple, then f must be an injection. 2. The Isomorphism Theorems 40 Direct Products Definition. If Hand K are groups, then their direct product, denoted by H x K, is the group with elements all ordered pairs (h, k), where h e Hand k e K, and with operation (h, k)(h', k') = (hh', kk'). It is easy to check that H x K is a group: the identity is (1, 1); the inverse (h, kti is (h- I , k- I ).