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 ebook. the 1st six chapters supply fabric for a primary direction, whereas the remainder of the e-book covers extra complex subject matters. This revised version keeps the readability of presentation that was once the hallmark of the former variations. From the experiences: "Rotman has given us a truly readable and worthwhile textual content, and has proven us many attractive vistas alongside his selected route." --MATHEMATICAL REVIEWS

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23. The commutator subgroup G' is a normal subgroup of G. Moreover, if H

If x E Sa, then x = sa for some s E S, and so x = srrb E Sb; similarly, if y E Sb, then y = s'b for some s' E S, and y = s' rr- 1 a E Sa. Therefore, Sa = Sb. 9. If S :-;::; G, then any two right (or any two left) co sets of Sin G are either identical or disjoint. Proof. We show that if there exists an element x E Sa n Sb, then Sa = Sb. Such an x has the form sb = x = ta, where s, t E S. Hence, ab- 1 = t- 1 s E S, and so the lemma gives Sa = Sb. 9 may be paraphrased to say that the right cosets of a subgroup S comprise a partition of G (each such coset is nonempty, and G is their disjoint union).

10 shows that there is no need to define a right index and a left index, for the number of right co sets is equal to the number ofleft cosets. It is a remarkable theorem of P. Hall (1935) that in a finite group G, one can always (as above) choose a common system ofrepresentatives for the right and left cosets of a subgroup S; if [G: S] = n, there exist elements t 1 , ••• , tn E G so that t 1 S, ... , tnS is the family of all left co sets and St 1 , ••• , Stn is the family of all right cosets. 2.

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