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

By Joseph J. Rotman

Fourth Edition

J.J. Rotman

An advent to the speculation of Groups

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

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Now 1 E Si for every i, and so 1 E n Si' If a, bEn Si, then a, b E Si for every i, and so ab- 1 E Si for every i; hence, ab- 1 E Si, and Si ~ G. 6. If X is a subset of a group G, then there is a smallest subgroup H of G containing X; that is, if XeS and S ~ G, then H ~ S. Proof. There are subgroups of G containing X; for example, G itself contains X; define H as the intersection of all the subgroups of G which contain X. 5, and X c H. If S ~ G and XeS, then S is one of the subgroups of G being intersected to form H; hence, H ~ S, and so H is the smallest such subgroup.

13, md = nk for some integer k, and b = am = (an/dt Therefore, (b) ~ (an/d), and this inclusion is equality because both subgroups have order d. 16. If n is a positive integer, then n = L

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.

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