You can verify that all the group axioms are satisfied for the operation defined above, but checking associativity directly can be rather time consuming. Fortunately, we have a theorem.

Let

be any group,

a cyclic group of order

and

an automorphism of

such that

is the identity automorphism. Then

is a group with respect to the operation

where

for all

and

is known as the cyclic extension of

by

induced by the automorphism

For a proof of the theorem (and more on cyclic extensions) see Chapter 21 of John F. Humphreys,

*A Course in Group Theory*, Oxford University Press, 1996. (Rock)

In our example,

is the mapping

which is an automorphism because

is Abelian. And

is not Abelian because

whereas

the two are not equal if

(Nerd)