Klein four-subgroup of 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) Klein four-group 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:Z4.
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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:

$\! H = \{ e, x, a^4, a^4x \}$

This is a subgroup of order four isomorphic to the Klein four-group, i.e., it is an elementary abelian group of prime-square order for the prime 2: all its non-identity elements have order 2.

Cosets

The subgroup has order 4 and index 4, so it has four cosets. Since it is a normal subgroup, the left cosets coincide with the right cosets:

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

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	4
index	4
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	$\langle a^2,x \rangle$	direct product of Z4 and Z2 in M16	direct product of Z4 and Z2
normal core	the subgroup itself	current page	Klein four-group
normal closure	the subgroup itself	current page	Klein four-group
characteristic core	the subgroup itself	current page	Klein four-group
characteristic closure	the subgroup itself	current page	Klein four-group
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
first omega subgroup	subgroup generated by the elements of order $p$, where $p$ is the underlying prime (in this case $p = 2$)	In fact, the elements $a^4, x, a^4x$ are the only elements of order two in M16, and they form a subgroup along with the identity element.
join of elementary abelian subgroups of maximum order	subgroup generated by all the elementary abelian subgroups of maximum order	In this case, it is the unique elementary abelian subgroup of maximum order.

Subgroup properties

Invariance under automorphisms and endomorphisms

Property	Meaning	Satisfied?	Explanation
normal subgroup	invariant under inner automorphisms	Yes	Precisely the set of elements of order two.
characteristic subgroup	invariant under all automorphisms	Yes	Precisely the set of elements of order two.
fully invariant subgroup	invariant under all endomorphisms	Yes	Precisely the set of elements of order two.
image-closed characteristic subgroup	image under any surjective homomorphism from whole group is characteristic in target group	No	Taking quotient by derived subgroup of M16, we get a direct factor of direct product of Z4 and Z2, which is hence not characteristic in it.
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 subgroup of elements of order at most two.
isomorph-normal subgroup	Every isomorphic subgroup is normal	Yes	Follows from being isomorph-free.
homomorph-containing subgroup	contains all homomorphic images	Yes	Any homomorphic image must comprise elements of order one or two, 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 1 or 2.
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