By Joseph J. Rotman
A person who has studied summary algebra and linear algebra as an undergraduate can comprehend this publication. the 1st six chapters offer fabric for a primary path, whereas the remainder of the publication covers extra complex subject matters. This revised version keeps the readability of presentation that was once the hallmark of the former versions. From the stories: "Rotman has given us a truly readable and helpful textual content, and has proven us many attractive vistas alongside his selected route." --MATHEMATICAL REVIEWS
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Extra info for An Introduction to the Theory of Groups (Graduate Texts in Mathematics, Volume 148)
Sample text
Now NaNb = Na(a-1Na)b = N(aa-1)Nab (because N is normal) = NNab = Nab (because N ~ G). Thus, NaNb = Nab, and so the product of two cosets is a coset. We let the reader prove that the identity is the coset N = N 1 and that the inverse of N a is N(a- 1 ). This group is denoted by GIN, and the definition of index gives IGINI = [G: N]. 22. , the function v: G -+ GIN = Na) is a surjective homomorphism with kernel N. Proof. The equation v(a)v(b) = v(ab) is just the formula NaNb = Nab; hence, v is a homomorphism.
If S :-;::; G, then the index of S in G, denoted by [G: S], is the number of right co sets of Sin G. 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, ...
Yff). For any group G, show that G' is the subset of all "long commutators": G' = {a l a 2 ••• anal-1 a 2-1 ••. an-1.. ai E G an d n > - 2} . M. Weichsel). 43. I. Cassidy (1979). (i) Let k[x, y] denote the ring of all polynomials in two variables over a field k, and let k[x] and k[y] denote the subrings of all polynomials in x and in y, respectively. Define G to be the set of all matrices of the form A= [ °1 1 °° J(x) h(x, y)] g(y) , 1 where J(x) E k[x], g(y) E key], and h(x, y) E k[x, y]. Prove that G is a multiplicative group and that G' consists of all those matrices for which J(x) = = g(y).