Derived subgroup of nontrivial semidirect product of Z4 and Z4

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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) 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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Definition

Consider the group:

G := \langle x,y \mid x^4 = y^4 = e, yxy^{-1} = x^3 \rangle.

This is a group of order 16 with elements:

\! \{ e,x,x^2,x^3,y,xy,x^2y,x^3y,y^2,xy^2,x^2y^2,x^3y^2,y^3,xy^3,x^2y^3,x^3y^3 \}

We are interested in the subgroup:

\! H = \{ e, x^2 \}

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:

\! \{ e, x^2 \}, \{ x , x^3 \}, \{ y, x^2y \}, \{ xy, x^3y \}, \{ y^2, x^2y^2 \}, \{xy^2, x^3y^2 \}, \{ y^3, x^2y^3 \}, \{ xy^3, x^3y^3 \}

Complements

The subgroup has no permutable complements. Since it is a normal subgroup, this also means it has no lattice complements.

Properties related to complementation

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
normal subgroup invariant under inner automorphisms Yes derived subgroup is normal
characteristic subgroup invariant under all automorphisms Yes derived subgroup is characteristic
fully invariant subgroup invariant under all endomorphisms Yes derived subgroup is fully invariant
verbal subgroup generated by set of words Yes derived subgroup is verbal
characteristic-isomorph-free subgroup no other isomorphic characteristic subgroup No \{ y^2, e \} and \{ x^2y^2, e\} are isomorphic characteristic subgroups. Hence also not a isomorph-free subgroup, isomorph-containing subgroup, or normal-isomorph-free subgroup.
isomorph-characteristic subgroup Every isomorphic subgroup is characteristic Yes The two other isomorphic subgroups are both characteristic, see central subgroup generated by a non-square in nontrivial semidirect product of Z4 and Z4 and subgroup generated by a non-commutator square in nontrivial semidirect product of Z4 and Z4.
isomorph-normal subgroup Every isomorphic subgroup is normal Yes Follows from being isomorph-characteristic.

Centrality and related properties

Property Meaning Satisfied? Explanation
central subgroup contained in the center Yes The center is \langle x^2, y^2 \rangle.
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]