Center 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:Z4 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 Klein four-group.
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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:

$\! H = \{ e, a^2, a^4, a^6 \}$

This subgroup is isomorphic to cyclic group:Z4. It is a normal subgroup and the quotient group is isomorphic to the Klein four-group.

Here is the multiplication table for $H$. Note that $H$ is an abelian group so we don't have to worry about left/right issues:

Element/element	$e$	$a^2$	$a^4$	$a^6$
$e$	$e$	$a^2$	$a^4$	$a^6$
$a^2$	$a^2$	$a^4$	$a^6$	$e$
$a^4$	$a^4$	$a^6$	$e$	$a^2$
$a^6$	$a^6$	$e$	$a^2$	$a^4$

Cosets

The subgroup is normal, so its left cosets coincide with its right cosets. Since it has order four and index $16/4 = 4$, there are four cosets:

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

The quotient group is isomorphic to a Klein four-group, and the multiplication table on cosets is given below. Note that the group is an abelian group, so we don't have to worry about left/right issues:

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

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	whole group $\langle a,x \rangle$		M16
normal core	the subgroup itself	current page	cyclic group:Z4
normal closure	the subgroup itself	current page	cyclic group:Z4
characteristic core	the subgroup itself	current page	cyclic group:Z4
characteristic closure	the subgroup itself	current page	cyclic group:Z4
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
center	elements that commute with every element of the group	Follows from multiplication table
ZJ-subgroup	center of the join of abelian subgroups of maximum order	The join of abelian subgroups of maximum order is the whole group, so its center is the group's center.
first agemo subgroup	in a finite $p$-group, subgroup generated by all $p^{th}$ powers. Here $p = 2$	This is precisely the set of squares in the group.

Subgroup properties

Invariance under automorphisms and endomorphisms: basic properties

Property	Meaning	Satisfied?	Explanation
normal subgroup	invariant under inner automorphisms	Yes	Follows from being the first agemo subgroup.
characteristic subgroup	invariant under all automorphisms	Yes	Follows from being the first agemo subgroup.
fully invariant subgroup	invariant under all endomorphisms	Yes	Follows from being the first agemo subgroup.

Word expression related properties and corollaries about invariance under automorphisms and endomorphisms

Property	Meaning	Satisfied?	Explanation
marginal subgroup	(complicated)	Yes	Follows from being the center. Center is marginal.
finite direct power-closed characteristic subgroup	any finite direct power is closed in the corresponding direct power of the center	Yes	center is finite direct power-closed characteristic, fully invariant implies finite direct power-closed characteristic
verbal subgroup	generated by set of words	Yes	Follows from being the first agemo 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.

Resemblance-based properties and corollaries about invariance under automorphisms and endomorphisms

Property	Meaning	Satisfied?	Explanation
isomorph-free subgroup	no other isomorphic subgroup	No	$\{e,a^2x,a^4,a^6x \}$ (see non-central Z4 in M16) is an isomorphic subgroup.
intermediately characteristic subgroup	characteristic in every intermediate subgroup	No	In the intermediate subgroup $\langle a^2, x \rangle$, which is direct product of Z4 and Z2 in M16, the subgroup is not characteristic.
isomorph-normal subgroup	every isomorphic subgroup is normal	Yes	The only other isomorphic subgroup is $\{e,a^2x,a^4,a^6x \}$ (see non-central Z4 in M16) and this is normal.
isomorph-characteristic subgroup	every isomorphic subgroup is characteristic	Yes	The only other isomorphic subgroup is $\{e,a^2x,a^4,a^6x \}$ (see non-central Z4 in M16) and this is characteristic.

Invariance under 1-automorphisms and 1-endomorphisms

Property	Meaning	Satisfied?	Explanation
1-endomorphism-invariant subgroup	invariant under all 1-endomorphisms of the group	Yes	Follows from its being precisely the set of squares.
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

Cohomology interpretation

We can think of $G$ as an extension with abelian normal subgroup $H$ and quotient group $G/H$. Since $H$ is in fact the center, the action of the quotient group on the normal subgroup is the trivial group action. We can thus study $G$ as an extension group arising from a cohomology class for the trivial group action of $G/H$ (which is a Klein four-group) on $H$ (which is cyclic group:Z4).

For more, see second cohomology group for trivial group action of V4 on Z4.

GAP implementation

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

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

The GAP display looks as follows:

gap> G := SmallGroup(16,6); H := Center(G);
<pc group of size 16 with 4 generators>
Group([ f3, f4 ])

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

gap> Order(G);
16
gap> Order(H);
4
gap> Index(G,H);
4
gap> H = Agemo(G,2,1);
true
gap> IsNormal(G,H);
true
gap> IsCharacteristicSubgroup(G,H);
true
gap> IsFullinvariant(G,H);
true