Center of M16

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:

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:

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

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.

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);  Group([ f3, f4 ])

Here is some GAP implementation to verify assertions made on this page:

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