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**Example text**

1 2). }; H(1 2 3) = {(1 2 3), (2 3)}; H(1 3 2) = {(1 3 2), (1 3)}. }; (1 2 3)H = {(1 2 3), (1 3)}; (1 3 2)H = {(1 3 2), (2 3)}. Notice that distinct right cosets are disjoint (as are distinct left cosets), just as in the example of parallel lines. Notice also that right cosets and left cosets can be distinct; for example, (1 2 3)H -# H(1 2 3); indeed, (1 2 3)H is not equal to any right coset of H in G. A right coset St has many representatives; every element of the form st for s E S is a representative of St.

N - 1, n). Prove that 8n can be generated by the two elements (1 2) and (1 2 ... n). Prove that 84 cannot be generated by (1 3) and (1 2 3 4). ) Lagrange's Theorem Definition. If S is a subgroup of G and if t E G, then a right coset of S in G is the subset of G St = {st: s E S} (a left coset is tS = {ts: s E S}). One calls t a representative of St (and also of tS). 3. Let G be the additive group of the plane 1R2: the elements of G are vectors (x, y), and addition is given by the "parallelogram law": (x, y) + (x', y') = (x + x', y + y').

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). (Hint. If A is denoted by (f, g, h), then (f, g, h)(f', g', h') = (f + 1', g + g', h + h' + Jg').