disprove: If G is a group and H, K < G are two subgroups, then (HK)≤ G.

Let a,b,c, and d be elements of a group G. Find an expression for (abcd)1 in terms of a1,b1,c1, and d1.

If H and K are arbitrary subgroups of G, prove that HK=KH if and only if HK is a subgroup of G.

Find the right regular representation of G as defined Exercise 11 for each of the following groups. a. G={ 1,i,1,i } from Example 1. b. The octic group D4={ e,,2,3,,,, }.

9. Find all homomorphic images of the octic group.

22. If and are both normal subgroups of , prove that is a normal subgroup of .

Label each of the following statements as either true or false. The Generalized Associative Law applies to any group, no matter what the group operation is.

True or False Label each of the following statements as either true or false. An element in a group may have more than one inverse.

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

34. Suppose that and are subgroups of the group . Prove that is a subgroup of .