Endomorphism structure of direct product of Z4 and Z2

This article discusses the structure of endomorphisms and in particular the automorphisms of the direct product of Z4 and Z2 -- the group obtained as the direct product of cyclic group:Z4 and cyclic group:Z2.

Description of endomorphism ring
We think of the group as the direct sum of $$\mathbb{Z}_4$$ and $$\mathbb{Z}_2$$, and we therefore get:

$$\operatorname{End}(\mathbb{Z}_4 \oplus \mathbb{Z}_2) = \operatorname{Hom}(\mathbb{Z}_4 \oplus \mathbb{Z}_2)$$

As an additive group, this is:

$$\operatorname{Hom}(\mathbb{Z}_4,\mathbb{Z}_4) \oplus \operatorname{Hom}(\mathbb{Z}_4,\mathbb{Z}_2) \oplus \operatorname{Hom}(\mathbb{Z}_2,\mathbb{Z}_4) \oplus \operatorname{Hom}(\mathbb{Z}_2,\mathbb{Z}_2) \cong \mathbb{Z}_4 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2<$$

The interpretation is as follows: any endomorphism of $$\mathbb{Z}_4 \oplus \mathbb{Z}_2$$ can be identified with a $$2 \times 2$$ matrix where the four entries are from the four direct summands on the right. Composition of endomorphisms works by matrix multiplication, where the entry-wise multiplication is by composition. This is part of general computational techniques for matrix endomorphisms.

Interpretation as wreath product of cyclic group of order two and cyclic group of order two
Note that the group has two cyclic subgroups of order four, which are automorphic subgroups and together generate the whole group. The automorphism group can now be viewed as follows:


 * Within either cyclic subgroup, we can interchange the two elements of order four keeping the other cyclic subgroup fixed. This gives an automorphism of order two. These two automorphisms of order two commute with each other and together generate a Klein four-group.
 * There is also an automorphism that interchanges the two cyclic subgroups and this automorphism acts by conjugation to interchange the two automorphisms mentioned above.
 * All automorphisms are generated by these, so the group is a semidirect product of a Klein four-group by a group of order two whose non-identity element interchanges the direct factors. This can also be thought of as the wreath product of cyclic group:Z2 and cyclic group:Z2 with the left-regular group action