Download Elementary Abstract Algebra by W. Edwin Clark PDF

By W. Edwin Clark

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Sample text

Guess the order of a k-cycle for arbitrary k. 10 Find the order of the following permutations: (a) (1 2)(3 4 5) (b) (1 2)(3 4)(5 6 7 8) (c) (1 2)(3 4)(5 6 7 8)(9 10 11) (d) Try to nd a rule for computing the order of a product disjoint cycles in terms of the sizes of the cycles. CHAPTER 4. 11 Find the order of each element of the group (Z6 +). 12 Find the order of each element of GL(2 Z2). Recall that GL(2 is the group of all 2 2 matrices with entries in Z2 with non-zero determinant. 13 Find the order of the element 2 in the group (R ; f0g ).

We will come back to this later. 2 (a) in the case that both G and H use additive notation, (b) in the case where G uses additive notation and H uses multiplicative notation, and (c) in the case where G uses multiplicative notation and H uses additive notation. 3 Let (G ) and (H ) be groups and let f : G ! H be an isomorphism. Then o(a) = o(f (a)) for all a 2 G. It follows that G and H have the same number of elements of each possible order. 3. 2. 4 If G and H are isomorphic groups, and G is abelian, then so is H .

The following exercise introduces a notation to make precise such matters. 19 (Challenge Problem) Let G be a group and let S G. De ne hS i to be the subset of G whose elements have the form s11 s22 snn where n 2 N , si 2 S and i = 1 for i = 1 2 : : : n. Prove ; ; ; ; ; 47 1. hS i is a subgroup of G. 2. hS i is the smallest subgroup of G that contains S , that is, if K is a subgroup of G and S K then hS i K . 3. Show that for n 3 the group Sn is not cyclic, but Sn = hf = (1 2) and = (1 2 n). gi where Note that the above problem shows that although Sn, n 3, is not cyclic, it is generated by two elements.

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