The group table of a group G is shown below. e a b C d b b a e h b f e e e C d a a ff b C 9 9 C d h h d a C d f g h df g h f d C b d C 9 g f e h h g f d The subset J = 9 h e a b C a a b с e b h a 9 b h e с d a e f f g {e, b} is a subgroup of G. (a) List all the distinct left cosets and all the distinct right cosets of J in G. (b) State whether J is normal in G, briefly justifying your answer.

Find the normalizer of the subgroup (1),(1,3)(2,4) of the octic group D4.

38. Let be the set of all matrices in that have the form with all three numbers , , and nonzero. Prove or disprove that is a group with respect to multiplication.

Prove that Ca=Ca1, where Ca is the centralizer of a in the group G.

23. Let be a group that has even order. Prove that there exists at least one element such that and . (Sec. ) Sec. 4.4, #30: 30. Let be an abelian group of order , where is odd. Use Lagrange’s Theorem to prove that contains exactly one element of order .

45. Let . Prove or disprove that is a group with respect to the operation of intersection. (Sec. )

In Exercises , is a normal subgroup of the group . Find the order of the quotient group . Write out the distinct elements of and construct a multiplication table for . 2. The octic group ; .

Let G=1,i,j,k be the quaternion group. List all cyclic subgroups of G. (Sec. 3.1,34).

If G is a cyclic group, prove that the equation x2=e has at most two distinct solutions in G.

17. Let and be elements of a group. Prove that is abelian if and only if .