Center of semidihedral group:SD16
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 (also denoted ) is the semidihedral group (also called quasidihedral group) of order . Specifically, it has the following presentation:
has 16 elements:
The subgroup of interest is the subgroup .
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.
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).
|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 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 can be central since it does not commute with . Among powers of , an element is central iff it equals its inverse. The only two such elements are and .||Definitional|
|third member of lower central series||commutator subgroup between whole group and its derived subgroup, i.e.,||Derived subgroup is , and taking the commutator again gives .||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 , here so subgroup generated by all elements of the form||Every fourth power is either or .||H = Agemo(G,2,2); using Agemo|
Invariance under automorphisms and endomorphisms
|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 .|
|isomorph-normal subgroup||Every isomorphic subgroup is normal||No||There are other subgroups of order two that are not normal: 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|