By Joseph J. Rotman

ISBN-10: 1461241766

ISBN-13: 9781461241768

ISBN-10: 1461286867

ISBN-13: 9781461286868

Fourth Edition

*J.J. Rotman*

*An creation to the speculation of Groups*

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

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Fourth EditionJ. J. RotmanAn advent to the speculation of Groups"Rotman has given us a truly readable and precious textual content, and has proven us many appealing vistas alongside his selected direction. "—MATHEMATICAL experiences

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**Additional resources for An Introduction to the Theory of Groups**

**Example text**

It follows that all the elements in the same conjugacy class have 3. Symmetric Groups and G-Sets 44 the same order. In particular, for any two elements x, Y E G, the elements xy and yx have the same order. If a EGis the sole resident of its conjugacy class, then a = gag- 1 for all g E G; that is, a commutes with every element of G. Definition. The center of a group G, denoted by Z(G), is the set of all a E G that commute with every element of G. It is easy to check that Z(G) is a normal abelian subgroup of G.

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 ..... H is a homomorphism, then the order of f(a) divides the order of a.

I) f(e) = e' , where e' is the identity in G'. , then f(a") = f(a)". Proof. (i) Applying f to the equation e = e * e gives f(e) = f(e * e) = f(e) 0 f(e). Now multiply each side of the equation by f(e)-l to obtain e' = f(e). (ii) Applyingfto the equations a * a-I = e = a-I * a gives f(a) 0 f(a- I ) = e' = f(a- I ) 0 f(a). 10, the uniqueness of the inverse, that f(a- I ) = f(a) -I. (iii) An easy induction proves f(a") = f(a)" for all n ~ 0, and then f(a-") = f«a- 1 )") = f(a- 1 )" = f(ar"· • Here are some examples.

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

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