By W. Edwin Clark
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Extra resources for Elementary Abstract Algebra
An is called the alternating group of degree n. 6 List the elements of An for n = 1 2 3 4. Based on this try to guess the order of An for n > 4. 2 Let a be an element of the group G. If there exists n 2 N such that an = e we say that a has nite order. and we de ne o(a) = minfn 2 N j an = eg If an 6= e for all n 2 N , we say that a has in nite order and we de ne o(a) = 1: In either case we call o(a) the order of a. Note carefully the di erence between the order of a group and the order of an element of a group.
3 (a) Determine which of the following subsets of S4 are subgroups of S4 . 1. H = f (1 2) (3 4) (1 2)(3 4)g 2. K = f (1 2 3) (1 3 2)g 3. J = f (1 2) (1 2 3)g 4. L = f 2 S4 j (1) = 1g. ) 1. A = f0 3 6 9 g 2. B = f0 6g Z12. 3. C = f0 1 2 3 4 5g (c) Determine which of the following subsets of Z are subgroups of Z. ) 1. U = f5k jk 2 Zg 2. V = f5k + 1 j k 2 N g 3. 4 Let SL(2 R ) = fA 2 GL(2 R) j det(A) = 1g: Prove that SL(2 R ) GL(2 R ). 5 For n 2 N , let An be the set of all even permutations in the group Sn .
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 : 13 11 7 6 5 4 3 10 2 12 14 1 15 9 8 Notice that the diagram consists of ve \cycles": one \6-cycle", one \4-cycle", two \2-cycles" and one \1-cycle". Every cycle diagram will look something like this. That's why we call it the cycle diagram. 3 Draw the cycle diagrams for all 24 elements of S4. You will need a systematic way to list the elements S4 to make sure you have not missed any. We now give a more precise de nition of a \cycle". 2 Let i1 i2 : : : ik be a list of k distinct elements from n].