Center of dihedral group:D16

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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:D16 (see subgroup structure of dihedral group:D16).
The subgroup is a normal subgroup and the quotient group is isomorphic to dihedral group:D8.
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Here, G is the dihedral group:D16, the dihedral group of order sixteen (and hence, degree eight). We use here the presentation:

G := \langle a,x \mid a^8 = x^2 = e, xax = a^{-1} \rangle

G has 16 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 \}

The subgroup H of interest is the subgroup \langle a^4 \rangle = \{ e, a^4 \}.

The quotient group is isomorphic to dihedral group:D8.

The subgroup is isomorphic to cyclic group:Z2 and the multiplication table is given below. Note that the group is an abelian group, so we do not need to worry about left/right issues:

Element/element e a^4
e e a^4
a^4 a^4 e

Cosets

The subgroup has order 2 and index 8, so it has 8 left cosets. It is a normal subgroup, so the left cosets coincide with the right cosets. The cosets are:

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

The quotient group is isomorphic to dihedral group:D8, and the multiplication table on cosets is given below. The row element is multiplied on the left and the column element is multiplied on the right.

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

Arithmetic functions

Function Value Explanation
order of whole group 16
order of subgroup 2
index of subgroup 8
size of conjugacy class (=index of normalizer) 1
number of conjugacy classes in automorphism class 1

Subgroup-defining functions

Subgroup-defining function What it means in general Why it takes this value GAP verification (set G := DihedralGroup(16); H := Center(G);) -- see more at #GAP implementation
center elements that commute with every group element No element of the form a^kx can be central since it does not commute with a. Among powers of a, an element is central iff it equals its inverse. The only two such elements are e and a^4. Definitional
third member of lower central series commutator subgroup between whole group and its derived subgroup, i.e., [[G,G],G] Derived subgroup is \langle a^2 \rangle, and taking the commutator again gives \langle a^4 \rangle. H = CommutatorSubgroup(G,DerivedSubgroup(G)); using CommutatorSubgroup and DerivedSubgroup
socle join of all minimal normal subgroups In fact, it is the unique minimal normal subgroup -- the group is a monolithic group. H = Socle(G); using Socle
second agemo subgroup subgroup generated by elements of the form g^{p^2}, here p = 2 so subgroup generated by all elements of the form g^4 Every fourth power is either e or a^4. H = Agemo(G,2,2); using Agemo

Subgroup properties

Invariance under automorphisms and endomorphisms

Property Meaning Satisfied? Explanation GAP verification (set G := DihedralGroup(16);H:= Center(G);) -- see more at #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 IsCharacteristicSubgroup(G,H); using IsCharacteristicSubgroup
fully invariant subgroup invariant under all endomorphisms Yes lower central series members are fully invariant, agemo subgroups are fully invariant IsFullinvariant(G,H); using IsFullinvariant
verbal subgroup generated by set of words Yes Lower central series members are 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, such as \{ x, e\}.
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 fourth powers, 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

GAP implementation

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

G := DihedralGroup(16); H := Center(G);

The GAP display looks as follows:

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

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

gap> Order(G);
16
gap> Order(H);
2
gap> Index(G,H);
8
gap> StructureDescription(G/H);
"D8"
gap> H = CommutatorSubgroup(G,DerivedSubgroup(G));
true
gap> H = Socle(G);
true
gap> H = Agemo(G,2,2);
true
gap> IsNormal(G,H);
true
gap> IsCharacteristicSubgroup(G,H);
true
gap> IsFullinvariant(G,H);
true