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:

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

and the center is the cyclic subgroup $C:=\{a^{2},e\}=\langle a^{2}\rangle$.

Subgroup-defining functions yielding this subgroup

• Center: The center of $G$ is $C$. To see that every element of $Z$ 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.
• Commutator subgroup: $C$ is the commutator subgroup. The quotient (called the abelianization) is $G/Z$, which is isomorphic to Klein four-group.
• Socle and monolith: In fact, this is the unique minimal normal subgroup.
• Frattini subgroup: $C$ is the intersection of the three maximal subgroups: $\langle a\rangle ,\langle a^{2},x\rangle ,\langle a^{2},ax\rangle$.
• first agemo subgroup: $\mho ^{1}(G)=C$. In other words, it is the subgroup generated by the squares.

Characteristicity and related properties satisfied

Property	Meaning	Satisfied?	Explanation
normal subgroup	invariant under inner automorphisms	Yes	center is normal
characteristic subgroup	invariant under all automorphisms	Yes	center is characteristic, commutator subgroup is characteristic
fully invariant subgroup	invariant under all endomorphisms	Yes	commutator subgroup is fully invariant, agemo subgroups are fully invariant
verbal subgroup	generated by set of words	Yes	commutator subgroup is verbal, agemo subgroups are verbal
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.

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