6. (i) Determine which of the following are groups under the stated operations. For those

which are groups, state the identity and inverses, and for those which are not, give

one axiom that fails.

(a) The set of 2 × 2 non-singular matrices, under addition,

(b) The set of positive real numbers, under division,

(c) {0, 2, 4, 6, 8), under addition modulo 10,

(d) {2, 4, 6, 8, 10, 12} under multiplication modulo 14.

(ii) Define a subgroup of a group G, and prove that the intersection of any two sub-

groups of G is a subgroup. (You may assume without proof that a subgroup has

the same identity as G, and the inverse of an element of the subgroup is the same

evaluated in G or the subgroup.) Give examples of subgroups H and K of (Z, +)

whose union is not a subgroup, and calculate what HK is.