Read e-book online An Introduction to the Theory of Groups (Graduate Texts in PDF

By Joseph J. Rotman

ISBN-10: 0387942858

ISBN-13: 9780387942858

Someone who has studied summary algebra and linear algebra as an undergraduate can comprehend this ebook. the 1st six chapters supply fabric for a primary direction, whereas the remainder of the booklet covers extra complex issues. This revised variation keeps the readability of presentation that was once the hallmark of the former versions. From the experiences: "Rotman has given us a really readable and useful 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 resources for An Introduction to the Theory of Groups (Graduate Texts in Mathematics, Volume 148)

Example text

Hence, gKg- 1 :::;; K for all g E G, and so K

The Isomorphism Theorems 30 if S, T, and U are nonempty subsets of G, then (ST) U = S(TU), for either side consists of all the elements of G of the form (st)u = s(tu) with s E S, t E T, and UE U. 20 (Product Formula). If Sand T are subgroups of a finite group G, then ISTIIS n TI = lSI I n Remark. The subset ST need not be a subgroup. 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}.

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 ..... H is a homomorphism, then the order of f(a) divides the order of a. 15. Prove that a group G of even order has an odd number of elements of order 2 (in particular, it has at least one such element). (Hint. 16. If H :5: G has index 2, then a2 E H for every a E G. 17. (i) If a, bEG commute and if am = 1 = bn , then (aW = 1, where k = lcm{m, n}.

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An Introduction to the Theory of Groups (Graduate Texts in Mathematics, Volume 148) by Joseph J. Rotman

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