Miami University Abstract Algebra Cosets and Laranges theorem Worksheet

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.