Center of semidihedral group:SD16

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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) semidihedral group:SD16 (see subgroup structure of semidihedral group:SD16).
The subgroup is a normal subgroup and the quotient group is isomorphic to dihedral group:D8.
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The semidihedral group SD_{16} (also denoted QD_{16}) is the semidihedral group (also called quasidihedral group) of order 16. Specifically, it has the following presentation:

G = SD_{16} := \langle a,x \mid a^8 = x^2 = e, xax^{-1} = a^3 \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 = \{ a^4, e \}.

The quotient group is isomorphic to dihedral group:D8.

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 \}

Note that the multiplication table for the quotient group looks identical to that for center of dihedral group:D16, although the original groups differ (semidihedral group:SD16 versus dihedral group:D16).

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 := SmallGroup(16,8); H := Center(G))
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 DerivedSubgroup and CommutatorSubgroup.
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
normal subgroup invariant under inner automorphisms Yes center is normal
characteristic subgroup invariant under all automorphisms Yes center is characteristic
fully invariant subgroup invariant under all endomorphisms Yes lower central series members are fully invariant, agemo subgroups are fully invariant
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