An Introduction to the Theory of Groups (Graduate Texts in by Joseph J. Rotman

By Joseph J. Rotman

Someone who has studied summary algebra and linear algebra as an undergraduate can comprehend this booklet. the 1st six chapters supply fabric for a primary path, whereas the remainder of the publication covers extra complex themes. This revised version keeps the readability of presentation that used to be the hallmark of the former variations. From the studies: "Rotman has given us a truly readable and priceless textual content, and has proven us many appealing vistas alongside his selected route." --MATHEMATICAL REVIEWS

Retail PDF from Springer; a number of chapters mixed into one dossier; scanned, searchable.

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Extra info for An Introduction to the Theory of Groups (Graduate Texts in Mathematics, Volume 148)

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12; multiplying by [a] now gives the desired result. 10. If G is a finite group and K :5: H :5: G, then [G: KJ = [G: HJ [H : KJ. 11. Let a E G have order n = mk, where m, k ~ 1. Prove that ak has order m. 12. (i) Prove that every group G of order 4 is isomorphic to either 14 or the 4-group V. (ii) If G is a group with IGI :5: 5, then G is abelian. 13. If a E G has order nand k is an integer with a k {k E 1: ak = 1} consists of all the multiplies of n. = 1, then n divides k. l4. If a E G has finite order and f: G .....

Because 71.. , and operation (a + (n») + (b + (n») = a + b + (n); in congruence class notation, [a] + [b] = [a + b]. /(n) is equal to 71.. n , the group of integers modulo n. An arbitrary quotient group GIN is often called G mod N because of this example. Definition. If a, bEG, the commutator 1 of a and b, denoted by [a, b], is [a, b] = aba- 1 b- 1 • The commutator subgroup (or derived subgroup) of G, denoted by G', is the subgroup of G generated by all the commutators. 43 below, that the subset of all commutators need not be a subgroup (the product of two commutators need not be a commutator).

13 shows that ker f =

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