- #1

Kate2010

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## Homework Statement

Let G1 and G2 be groups, let G = G1 x G2 and define the binary operation on G by

(a1,a2)(b1,b2):=(a1b1,a2b2)

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.

## Homework Equations

## 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.

Thanks :)