Derived 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) cyclic group: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 direct product of Z4 and 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:

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

We are interested in the subgroup:

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

This is a cyclic subgroup of order two, i.e., it is isomorphic to cyclic group:Z2. The quotient group is isomorphic to the abelian group direct product of Z4 and Z2.

Cosets

The subgroup is a normal subgroup and hence its left cosets coincide with its right cosets. The eight cosets are given below:

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

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	2
index	8
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	whole group $\langle a,x\rangle$		M16
normal core	the subgroup itself	current page	cyclic group:Z2
normal closure	the subgroup itself	current page	cyclic group:Z2
characteristic core	the subgroup itself	current page	cyclic group:Z2
characteristic closure	the subgroup itself	current page	cyclic group:Z2
commutator with whole group	trivial subgroup	--	trivial group

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
derived subgroup	subgroup generated by all commutators between elements of the group	The only commutators are the identity element and $a^{4}$.
socle	subgroup generated by the minimal normal subgroups. For a group of prime power order $G$, this is the same as $\Omega _{1}(Z(G))$ -- the identity element and all elements of prime order in the center	It is the unique minimal normal subgroup. The center is in fact $\langle a^{2}\rangle$, and this is precisely the set of elements of order at most 2 in the center.
second agemo subgroup	subgroup generated by all $(p^{2})^{th}$ powers where $p$ is the underlying prime (here $p=2$)	The fourth powers are precisely the identity and $a^{4}$.

Subgroup properties

Invariance under automorphisms and endomorphisms

Property	Meaning	Satisfied?	Explanation
normal subgroup	invariant under inner automorphisms	Yes	Follows from its being the derived subgroup.
characteristic subgroup	invariant under all automorphisms	Yes	Follows from its being the derived subgroup.
fully invariant subgroup	invariant under all endomorphisms	Yes	Follows from its being the derived subgroup.
verbal subgroup	generated by set of words	Yes	Follows from its being the derived subgroup.
image-closed characteristic subgroup	image under any surjective homomorphism from whole group is characteristic in target group	Yes	Follows from its being a verbal subgroup.
image-closed fully invariant subgroup	image under any surjective homomorphism from whole group is fully invariant in target group	Yes	Follows from its being a verbal subgroup.
isomorph-free subgroup	no other isomorphic subgroup	No	$\{x,e\}$ is another isomorphic subgroup.
isomorph-normal subgroup	every isomorphic subgroup is normal	No	$\{x,e\}$ is a non-normal isomorphic subgroup.
normal-isomorph-free subgroup	no other isomorphic normal subgroup	Yes	Follows from its being the socle.
homomorph-containing subgroup	contains all homomorphic images	No	Follows from not being isomorph-free.
1-endomorphism-invariant subgroup	invariant under all 1-endomorphisms of the group	Yes	Follows from its being precisely the set of fourth powers.
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

GAP implementation

The group and subgroup can be constructed using GAP's SmallGroup and DerivedSubgroup functions as follows:

G := SmallGroup(16,6); H := DerivedSubgroup(G);

The GAP display looks as follows:

gap> G := SmallGroup(16,6); H := DerivedSubgroup(G);
<pc group of size 16 with 4 generators>
Group([ f4 ])

Here is some GAP implementation to verify assertions made on this page: [SHOW MORE]

gap> Order(G);
16
gap> Order(H);
2
gap> Index(G,H);
8
gap> IsNormal(G,H);
true
gap> IsCharacteristicSubgroup(G,H);
true
gap> IsFullinvariant(G,H);
true