Let G and H be groups with a elementof G and b elementof H F

Let G and H be groups, with a elementof G and b elementof H. Find two cyclic groups G and H such that G times H is not cyclic.

Solution

There appear to be two different questions. The 1st question is incomplete.

2. Let G and H be two finite cyclic groups with orders p and q respectively and let G = <g> and H = <h> . Then,|GxH |= |G||H| =pq so that GxH is cyclic if and only if GxH has an element of order pq.

Now, let g^m and hⁿ be 2 arbitrary elements of G and H respectively, where 0 m < p and 0 n < q. Then (g^m, hⁿ) G x H. Further, the order of (g^m, hⁿ) can be pq if and only if , both m and n are divisors of pq i.e. pq = am = bn, where a and b are integers. Now, if p and q are relatively prime, then the order of (g^m, hⁿ) cannot be pq. Further, since (g^m, hⁿ) is an arbitrary element of GxH, hence, when the orders of the cyclic groups G and H are relatively prime, then GxH cannot be cyclic.