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## Miami University Abstract Algebra Cosets and Laranges theorem Worksheet

### Question Description

1. List the left cosets of the subgroups in each of the following. Here ?a? denotes the subgroupgenerated by the element a.

(a) ?8? in (Z24, +)
(b) ?3? in U(8)
(c)3ZinZ (where3Z={3k:k?Z}.)
(d) An in Sn (where An is the set of all even permutations on {1, . . . , n}.)

2. Find all the left cosets of H = {1, 19} in U (30).

3. Given a finite group G and H a subgroup. Using the same argument as in the Lemma provedin class, one can show all the statements of the lemma hold analogously for right cosets of H in G.In particular, Ha = H if and only if a ? H and the distinct right cosets of H form a partition ofG. Now, suppose that |H| = |G|/2.

(a) Show that for every a ? G, aH = Ha. (comment: in general aH and Ha are not necessarilyequal. But with our condition |H| = |G|/2 here, this indeed holds.)

(b) Suppose a,b ? G are two elements of G that are not in H. Prove that ab ? H.4. Let G be a group of order 63. Prove that G must have an element of order 3.

5. Let G be a group of order 155. Suppose a, b are two nonidentity elements of G that have differentorders. Prove that the only subgroup of G that contains both a and b must be G itself. (hint: ByLaranges theorem, the order of any nonidentity element must be one of 5,31,155. If one of a,bhas order 155 then the statement is quite easy to prove. So one may assume |a| = 5 and |b| = 31.Consider how Theorem 7.2 might be relevant.)

6. (Graduates only) Prove that every subgroup of Dn that has an odd order must be cyclic.

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