Center of dihedral group:D8

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) dihedral group:D8 (see subgroup structure of dihedral group:D8).
The subgroup is a normal subgroup and the quotient group is isomorphic to Klein four-group.
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This article discuss the dihedral group of order eight and its center, which is a cyclic group of order two.

The dihedral group of order eight is defined as:

$G := \langle a,x \mid a^4 = x^2 = e, xax = a^{-1} \rangle$.

It has multiplication table:

The row element is multiplied on the left and the column element is multiplied on the right.

Element	$\! e$	$\! a$	$\! a^2$	$\! a^3$	$\! x$	$\! ax$	$\! a^2x$	$\! a^3x$
$\! e$	$\! e$	$\! a$	$\! a^2$	$\! a^3$	$\! x$	$\! ax$	$\! a^2x$	$\! a^3x$
$\! a$	$\! a$	$\! a^2$	$\! a^3$	$\! e$	$\! ax$	$\! a^2x$	$\! a^3x$	$\! x$
$\! a^2$	$\! a^2$	$\! a^3$	$\! e$	$\! a$	$\! a^2x$	$\! a^3x$	$\! x$	$\! ax$
$\! a^3$	$\! a^3$	$\! e$	$\! a$	$\! a^2$	$\! a^3x$	$\! x$	$\! ax$	$\! a^2x$
$\! x$	$\! x$	$\! a^3x$	$\! a^2x$	$\! ax$	$\! e$	$\! a^3$	$\! a^2$	$\! a$
$\! ax$	$\! ax$	$\! x$	$\! a^3x$	$\! a^2x$	$\! a$	$\! e$	$\! a^3$	$\! a^2$
$\! a^2x$	$\! a^2x$	$\! ax$	$\! x$	$\! a^3x$	$\! a^2$	$\! a$	$\! e$	$\! a^3$
$\! a^3x$	$\! a^3x$	$\! a^2x$	$\! ax$	$\! x$	$\! a^3$	$\! a^2$	$\! a$	$\! e$

and the center is the cyclic subgroup:

$H := \{ a^2, e \} = \langle a^2 \rangle$.

It has multiplication table:

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

Cosets

The subgroup has the following four cosets:

$\! \{ e, a^2 \}, \qquad \{ x, a^2x \}, \qquad \{ a, a^3 \}, \qquad \{ ax, a^3x \}$

The quotient group is isomorphic to Klein four-group, and the multiplication table is as follows:

Element/element	$\{ e, a^2 \}$	$\{ a, a^3 \}$	$\{ x, a^2x \}$	$\{ ax, a^3x \}$
$\{ e, a^2 \}$	$\{ e, a^2 \}$	$\{ a, a^3 \}$	$\{ x, a^2x \}$	$\{ ax, a^3x \}$
$\{ a, a^3 \}$	$\{ a, a^3 \}$	$\{ e, a^2 \}$	$\{ ax, a^3x \}$	$\{ x, a^2x \}$
$\{ x, a^2x \}$	$\{ x,a^2x \}$	$\{ ax, a^3x \}$	$\{ e, a^2 \}$	$\{ a, a^3 \}$
$\{ ax, a^3x \}$	$\{ax, a^3x \}$	$\{ x, a^2x \}$	$\{ a, a^3 \}$	$\{ e, a^2 \}$

Complements

The subgroup $H$ has no permutable complement and also has no lattice complement.

Properties related to complementation

Property	Meaning	Satisfied?	Explanation
complemented normal subgroup	normal subgroup with a complement.	No	nilpotent and non-abelian implies center is not complemented
permutably complemented subgroup	subgroup with a permutable complement.	No	nilpotent and non-abelian implies center is not complemented
lattice-complemented subgroup	subgroup with a lattice complement.	No	nilpotent and non-abelian implies center is not complemented

Arithmetic functions

Function	Value	Explanation
order of whole group	8
order of subgroup	2
index of subgroup	4
size of conjugacy class of subgroup (=index of normalizer)	1	center is normal, so the conjugacy class has size 1
number of conjugacy classes in automorphism class of subgroup	1	center is characteristic
size of automorphism class of subgroup	1	center is characteristic

Effect of subgroup operators

Function	Value as subgroup (descriptive)	Value as subgroup (link)	Value as group
normalizer	the whole group	--	dihedral group:D8
centralizer	the whole group	current page	dihedral group:D8
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	the trivial subgroup	current page	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	Corresponding quotient-defining function	GAP verification (set G := DihedralGroup(8); H := Center(G);) -- see more at #GAP implementation
center	set of elements that commute with every element	To see that every element of $H$ is in the center, note that $a^2$ commutes with both $a$ and $x$. To see that no other element is in the center, note that $a$ and $x$ do not commute.	inner automorphism group	Definitional
derived subgroup	subgroup generated by commutators of all pairs of elements in the group, smallest subgroup with abelian quotient	The quotient (called the abelianization) is $G/H$, which is isomorphic to Klein four-group. No smaller subgroup works, because the quotient by the trivial subgroup is isomorphic to $G$, which is non-abelian.	abelianization	H = DerivedSubgroup(G); using DerivedSubgroup
socle	subgroup generated by all the minimal normal subgroups	It is the unique minimal normal subgroup (hence is also a monolith). In general, for a finite $p$-group, the socle is $\Omega_1(Z(G))$. See socle equals Omega-1 of center for nilpotent p-group.		H = Socle(G); using Socle
Frattini subgroup	intersection of all the maximal subgroups	$H$ is the intersection of $\langle a \rangle, \langle a^2, x \rangle, \langle a^2, ax \rangle$.	Frattini quotient	H = FrattiniSubgroup(G); using FrattiniSubgroup
Jacobson radical	intersection of all the maximal normal subgroups	For a group of prime power order, this is same as the Frattini subgroup, because nilpotent implies every maximal subgroup is normal.
Baer norm	intersection of normalizers of all subgroups	Normalizer of $\langle x \rangle$ and of $\langle a^2x \rangle$ is $\langle a^2,x \rangle$. Normalizer of $\langle ax \rangle$ and of $\langle a^3x \rangle$ is $\langle a^2, ax \rangle$. All other normalizers are the whole group. Intersection of the two proper normalizers is $\langle a^2 \rangle = \{ e, a^2\}$.
first agemo subgroup	subgroup generated by all $p^{th}$ powers, where $p$ is the underlying prime (in this case 2)	$\mho^1(G) = H$. In other words, it is the subgroup generated by the squares. In this case, it is precisely' the set of squares.		H = Agemo(G,2,1); using Agemo
ZJ-subgroup	center of the join of abelian subgroups of maximum order	The join of abelian subgroups of maximum order (the Thompson subgroup) is the whole group dihedral group:D8, so its center is $H$.
D*-subgroup	abelian and any element acting quadratically on it acts linearly on it (roughly speaking)	In a group of nilpotency class two, this subgroup coincides with the center

Subgroup properties

Invariance under automorphisms and endomorphisms

Property	Meaning	Satisfied?	Explanation	GAP verification (set G := DihedralGroup(8); H := Center(G);) -- see #GAP implementation
normal subgroup	invariant under inner automorphisms	Yes	center is normal	IsNormal(G,H); using IsNormal
characteristic subgroup	invariant under all automorphisms	Yes	center is characteristic, derived subgroup is characteristic	IsCharacteristicSubgroup(G,H); using IsCharacteristicSubgroup
fully invariant subgroup	invariant under all endomorphisms	Yes	derived subgroup is fully invariant, agemo subgroups are fully invariant	IsFullinvariant(G,H); using IsFullinvariant
verbal subgroup	generated by set of words	Yes	derived subgroup is verbal, agemo subgroups are verbal
marginal subgroup		Yes	center is marginal
normal-isomorph-free subgroup	no other isomorphic normal subgroup	Yes
isomorph-free subgroup, isomorph-containing subgroup	No other isomorphic subgroups	No	There are other subgroups of order two.
isomorph-normal subgroup	Every isomorphic subgroup is normal	No	There are other subgroups of order two that are not normal: $\langle x \rangle$ etc.
homomorph-containing subgroup	contains all homomorphic images	No	There are other subgroups of order two.
1-endomorphism-invariant subgroup	invariant under all 1-endomorphisms of the group	Yes	It is precisely the set of squares, which must therefore go to squares under 1-endomorphisms
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

Centrality and related properties

Property	Meaning	Satisfied?	Explanation
central subgroup	contained in the center	Yes	In fact, it is equal to the center.
central factor		Yes	(because it is central).
transitively normal subgroup		Yes	(because it is a central factor).
SCAB-subgroup		Yes

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:Z2).

For more, see second cohomology group for trivial group action of V4 on Z2.

GAP implementation

The group and subgroup pair can be defined using GAP's DihedralGroup and Center functions as follows:

G := DihedralGroup(8); H := Center(G);

The GAP display looks as follows:

gap> G := DihedralGroup(8); H := Center(G);
<pc group of size 8 with 3 generators>
Group([ f3 ])