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

Retail PDF from Springer; a number of chapters mixed into one dossier; scanned, searchable.

**Read Online or Download An Introduction to the Theory of Groups (Graduate Texts in Mathematics, Volume 148) PDF**

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**Additional resources for An Introduction to the Theory of Groups (Graduate Texts in Mathematics, Volume 148)**

**Example text**

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.