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

By Joseph J. Rotman

Somebody who has studied summary algebra and linear algebra as an undergraduate can comprehend this publication. the 1st six chapters offer fabric for a primary direction, whereas the remainder of the ebook covers extra complex subject matters. This revised variation keeps the readability of presentation that was once the hallmark of the former versions. From the stories: "Rotman has given us a really readable and worthwhile textual content, and has proven us many attractive 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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Prove that if G is an abelian group of prime exponent p, then G is a vector space over 7i. p , and every homomorphism rp : G -+ G is a linear transformation. Moreover, a finite abelian p-group G is elementary if and only if it has exponentp. CHAPTER 3 Symmetric Groups and G-Sets The definition of group arose from fundamental properties of the symmetric group Sn· But there is another important feature of Sn: its elements are functions acting on some underlying set, and this aspect is not explicit in our presentation so far.

Is this true if we replace "two subgroups" by "three subgroups"? 4. Let S be a proper subgroup of G. If G - S is the complement of S, prove that (G - S) = G. 5. Let f: G -+ Hand g: G -+ H be homomorphisms, and let K = {a E G: f(a) = g(a)}. Must K be a subgroup of G? 6. Suppose that X is a nonempty subset of a set Y. 7. If n > 2, then An is generated by all the 3-cycles. (Hint. 8. + 2 , but show, for n ~ 2, that S. 9. 2. The Isomorphism Theorems (i) (ii) (iii) (iv) Prove that 8n can be generated by (1 2), (1 3), ...

We usually write kerf instead of kernel f and imf instead of image f. We have been using multiplicative notation, but it is worth writing the definition of subgroup in additive notation as well. If G is an additive group, then a nonempty subset S of G is a subgroup of G if s E S implies - s E Sand s, t E Simply s + t E S. 2 says that S is a subgroup if and only if oE Sand s, t E Simply s - t E S. 5. The intersection of any family of subgroups of a group G is again a subgroup of G. Proof. Let {Si: i E I} be a family of subgroups of G.

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