Cyclic maximal subgroup of semidihedral group:SD16
From Groupprops
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:Z8 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 cyclic group:Z2.
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Here, is the semidihedral group:SD16, the semidihedral group of order sixteen (and hence, degree eight). We use here the presentation:
has 16 elements:
The subgroup of interest is the subgroup
. It is cyclic of order 8 and is given by:
Contents
Cosets
The subgroup has index two and is hence normal (since index two implies normal). Its left cosets coincide with its right cosets, and there are two cosets:
Complements
COMPLEMENTS TO NORMAL SUBGROUP: TERMS/FACTS TO CHECK AGAINST:
TERMS: permutable complements | permutably complemented subgroup | lattice-complemented subgroup | complemented normal subgroup (normal subgroup that has permutable complement, equivalently, that has lattice complement) | retract (subgroup having a normal complement)
FACTS: complement to normal subgroup is isomorphic to quotient | complements to abelian normal subgroup are automorphic | complements to normal subgroup need not be automorphic | Schur-Zassenhaus theorem (two parts: normal Hall implies permutably complemented and Hall retract implies order-conjugate)
There are four possible permutable complements to in
, all of them automorphic to each other:
Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|
complemented normal subgroup | normal subgroup with permutable complement | Yes | see above | |
permutably complemented subgroup | subgroup with permutable complement | Yes | (via complemented normal) | |
lattice-complemented subgroup | subgroup with lattice complement | Yes | (via permutably complemented) | |
retract | has a normal complement | No | ||
direct factor | normal subgroup with normal complement | No |
Arithmetic functions
Function | Value | Explanation |
---|---|---|
order of whole group | 16 | |
order of subgroup | 8 | |
index of subgroup | 2 | |
size of conjugacy class = index of normalizer | 1 | |
number of conjugacy classes in automorphism class | 1 |
Effect of subgroup operators
Function | Value as subgroup (descriptive) | Value as subgroup (link) | Value as group |
---|---|---|---|
normalizer | whole group | -- | semidihedral group:D16 |
centralizer | ![]() |
current page | cyclic group:Z8 |
normal core | the subgroup itself | current page | cyclic group:Z8 |
normal closure | the subgroup itself | current page | cyclic group:Z8 |
characteristic core | the subgroup itself | current page | cyclic group:Z8 |
characteristic closure | the subgroup itself | current page | cyclic group:Z8 |
Subgroup-defining functions
Subgroup-defining function | Meaning in general | Why it takes this value |
---|---|---|
centralizer of derived subgroup | centralizer of the derived subgroup (the commutator of the group with itself) | The derived subgroup is ![]() ![]() |
Subgroup properties
Invariance under automorphisms and endomorphisms
Property | Meaning | Satisfied? | Explanation |
---|---|---|---|
normal subgroup | invariant under inner automorphisms | Yes | |
characteristic subgroup | invariant under all automorphisms | Yes | On account of being the centralizer of derived subgroup, also on account of being an isomorph-free subgroup. |
fully invariant subgroup | invariant under all endomorphisms | No | not invariant under the retraction with kernel ![]() ![]() |
isomorph-free subgroup | no other isomorphic subgroup | Yes |