Derived subgroup of nontrivial semidirect product of Z4 and Z4
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:Z2 and the group is (up to isomorphism) nontrivial semidirect product of Z4 and Z4 (see subgroup structure of nontrivial semidirect product of Z4 and Z4).
The subgroup is a normal subgroup and the quotient group is isomorphic to direct product of Z4 and Z2.
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Contents
Definition
Consider the group:
.
This is a group of order 16 with elements:
We are interested in the subgroup:
This is the derived subgroup. In particular, it is a normal subgroup and the quotient group is isomorphic to direct product of Z4 and Z2.
Cosets
The subgroup is a normal subgroup, so its left cosets coincide with its right cosets. The subgroup has order 2 and index 8, so there are 8 cosets, given as:
Complements
The subgroup has no permutable complements. Since it is a normal subgroup, this also means it has no lattice complements.
Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|
complemented normal subgroup | normal subgroup with permutable complement | No | see above | |
permutably complemented subgroup | subgroup with permutable complement | No | ||
lattice-complemented subgroup | subgroup with lattice complement | No | ||
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 | 2 | |
index | 8 | |
size of conjugacy class | 1 | |
number of conjugacy classes in automorphism class | 1 |
Subgroup-defining functions
Subgroup-defining function | Meaning in general | Why it takes this value |
---|---|---|
derived subgroup | subgroup generated by all commutators | The commutators are precisely the elements of this subgroup. |
Subgroup properties
Invariance under automorphisms and endomorphisms
Property | Meaning | Satisfied? | Explanation |
---|---|---|---|
central subgroup | contained in the center | Yes | The center is ![]() |
central factor | product with centralizer is whole group | Yes | central implies central factor |
GAP implementation
The group and subgroup can be constructed as follows, using the SmallGroup and DerivedSubgroup functions:
G := SmallGroup(16,4); H := DerivedSubgroup(G);
Implementing this in GAP looks as follows:
gap> G := SmallGroup(16,4); H := DerivedSubgroup(G); <pc group of size 16 with 4 generators> Group([ f3 ])Below is GAP implementation to test some of the assertions in this page:[SHOW MORE]