Let G1 and G2 be groups, let G = G1 x G2 and define the binary operation on G by
Prove that this makes G into a group. Prove G is abelian iff G1 and G2 are abelian.
Hence or otherwise give examples of a non-cyclic abelian group of order 8 and a non-abelian group of order 42.
The Attempt at a Solution
I have done the 1st part of this question and I'm just struggling with the examples. From reading around the subject I think a non-cyclic abelian group of order 8 would be Z2 x Z2 x Z2 where Z2 is the integers modulo 2 under addition. However, I don't really understand this.
Also, I'm unsure how to tackle the non-abelian group of order 42.