Endomorphism structure of direct product of Z4 and Z2

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This article gives specific information, namely, endomorphism structure, about a particular group, namely: direct product of Z4 and Z2.
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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.

Summary of information

Construct Value Order Second part of GAP ID (if applicable)
endomorphism ring The additive group is isomorphic to direct product of E8 and Z4 32 45 (for additive group)
automorphism group dihedral group:D8 8 3
inner automorphism group trivial group 1 1
extended automorphism group dihedral group:D8 8 3
quasiautomorphism group dihedral group:D8 8 3
1-automorphism group direct product of D8 and Z2 16 11
outer automorphism group dihedral group:D8 8 3

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. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

Description of automorphism group

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