Center of dihedral group:D8

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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: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
D8latticeofsubgroups.png

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 ])