Direct product of Z4 and Z2 in M16

This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) direct product of Z4 and Z2 and the group is (up to isomorphism) M16 (see subgroup structure of M16).
The subgroup is a normal subgroup and the quotient group is isomorphic to cyclic group:Z2.
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Definition

We consider the group:

$G = M_{16} = \langle a,x \mid a^8 = x^2 = e, xax = a^5 \rangle$

with $e$ denoting the identity element.

This is a group of order 16, with elements:

$\{ e, a, a^2, a^3, a^4, a^5, a^6, a^7, x, ax, a^2x, a^3x, a^4x, a^5x, a^6x, a^7x \}$

We are interested in the subgroup:

$H = \{ e, a^2, a^4, a^6, x, a^2x, a^4x, a^6x \} = \langle a^2,x \rangle$

This is a subgroup of order eight isomorphic to direct product of Z4 and Z2, where the cyclic four-subgroup is $\langle a^2 \rangle$ and the cyclic two-subgroup is $\langle x \rangle$.

Cosets

The subgroup is a subgroup of index two, hence it is a normal subgroup (see index two implies normal) and in particular its left cosets coincide with its right cosets. The two cosets are:

$H =\{ e, a^2, a^4, a^6, x, a^2x, a^4x, a^6x \}, G \setminus H = \{ a, a^3, a^5, a^7, ax, a^3x, a^5x, a^7x \}$

Complements

The subgroup has no permutable complements. Since it is a normal subgroup, this also means it has no lattice complements.

Properties related to complementation

Property	Meaning	Satisfied?	Explanation	Comment
complemented normal subgroup	normal subgroup with permutable complement	No	see above
permutably complemented subgroup	subgroup with permutable complement	No
lattice-complemented subgroup	subgroup with lattice complement	No
retract	has a normal complement	No
direct factor	normal subgroup with normal complement	No

Arithmetic functions

Function	Value	Explanation
order of whole group	16
order of subgroup	8
index	2
size of conjugacy class	1
number of conjugacy classes in automorphism class	1

Effect of subgroup operators

Function	Value as subgroup (descriptive)	Value as subgroup (link)	Value as group
normalizer	whole group $\langle a,x \rangle$		M16
centralizer	the subgroup itself	current page	direct product of Z4 and Z2
normal core	the subgroup itself	current page	direct product of Z4 and Z2
normal closure	the subgroup itself	current page	direct product of Z4 and Z2
characteristic core	the subgroup itself	current page	direct product of Z4 and Z2
characteristic closure	the subgroup itself	current page	direct product of Z4 and Z2
commutator with whole group	$\langle a^4 \rangle$	derived subgroup of M16	cyclic group:Z2

Subgroup-defining functions

The subgroup is a characteristic subgroup of the whole group and arises as a result of many subgroup-defining functions on the whole group. Some of these are given below.

Subgroup-defining function	Meaning in general	Why it takes this value
second omega subgroup	For a group of prime power order, subgroup generated by all elements whose order divides the square of the prime	All elements in the subgroup have order at most 4, all elements outside it have order 8.
join of abelian subgroups of maximum rank	For a group of prime power order, this is the subgroup generated by all the abelian subgroups of maximum rank.	The maximum possible rank of an abelian subgroup is 2. There are two such subgroups, V4 in M16 and this subgroup, and this is the bigger of the two.

Subgroup properties

Invariance under automorphisms and endomorphisms

Property	Meaning	Satisfied?	Explanation
normal subgroup	invariant under inner automorphisms	Yes	Index two implies normal
characteristic subgroup	invariant under all automorphisms	Yes	Follows from being the largest abelian subgroup of rank two.
fully invariant subgroup	invariant under all endomorphisms	Yes	Follows from its being the second omega subgroup -- all elements in the subgroup have order at most four, all elements outside it have order bigger than four.
image-closed characteristic subgroup	image under any surjective homomorphism from whole group is characteristic in target group	No	Taking quotient by center of M16, we get Z2 in V4, which is not characteristic.
image-closed fully invariant subgroup	image under any surjective homomorphism from whole group is fully invariant in target group	No	Follows from not being image-closed characteristic.
verbal subgroup	generated by set of words	No	Follows from its not being an image-closed characteristic subgroup
isomorph-free subgroup	no other isomorphic subgroup	Yes	It is precisely the set of elements of order at most four.
isomorph-normal subgroup	all isomorphic subgroups are normal in the whole group	Yes	Follows from being isomorph-free
homomorph-containing subgroup	contains all homomorphic images	Yes	Any homomorphic image must comprise elements of order at most four, all of which are in this subgroup.
1-endomorphism-invariant subgroup	invariant under all 1-endomorphisms of the group	Yes	Under any 1-endomorphism, all elements must go to elements of order at most 4.
1-automorphism-invariant subgroup	invariant under all 1-automorphisms of the group	Yes	Follows from being 1-endomorphism-invariant.
quasiautomorphism-invariant subgroup	invariant under all quasiautomorphisms	Yes	Follows from being 1-automorphism-invariant