Central subgroup generated by a non-square in nontrivial semidirect product of Z4 and Z4

From Groupprops
Jump to: navigation, search
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 quaternion group.
VIEW: Group-subgroup pairs with the same subgroup part | Group-subgroup pairs with the same group part| Group-subgroup pairs with the same quotient part | All pages on particular subgroups in groups

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^2y^2 \}

The quotient group is isomorphic to quaternion group.

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

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 properties

Invariance under automorphisms and endomorphisms

Property Meaning Satisfied? Explanation
normal subgroup invariant under inner automorphisms Yes central implies normal
characteristic subgroup invariant under all automorphisms Yes The non-identity element is the unique element in the center that is not a square.
fully invariant subgroup invariant under all endomorphisms No The endomorphism x \mapsto e, y \mapsto x sends this subgroup to \{ e, x^2 \}, the derived subgroup of nontrivial semidirect product of Z4 and Z4.

GAP implementation

The group and subgroup pair can be constructed using GAP as follows:

G := SmallGroup(16,4); H := Group(Difference(Set(Center(G)),Set(List(G,x -> x^2))));

The GAP display looks as below:

gap> G := SmallGroup(16,4); H := Group(Difference(Set(Center(G)),Set(List(G,x -> x^2))));
<pc group of size 16 with 4 generators>
<pc group with 1 generators>

Here is some GAP code verifying some of the assertions on this page:

gap> Order(G);
16
gap> Order(H);
2
gap> Index(G,H);
8
gap> StructureDescription(H);
"C2"
gap> StructureDescription(G/H);
"Q8"
gap> IsNormal(G,H);
true
gap> IsCharacteristicSubgroup(G,H);
true
gap> IsFullinvariant(G,H);
false