Z4 in direct product of Z4 and Z2

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:Z4 and the group is (up to isomorphism) direct product of Z4 and Z2 (see subgroup structure of direct product of Z4 and Z2).
The subgroup is a normal subgroup and the quotient group is isomorphic to cyclic group:Z2.
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

The group G is a direct product of Z4 and Z2, which, for convenience, we describe by ordered pairs with the first members from the integers mod 4 (the first direct factor cyclic group:Z4) and the second member from the integers mod 2 (the second direct factor cyclic group:Z2). It has elements:

\! \{ (0,0), (1,0), (2,0), (3,0), (0,1), (1,1), (2,1), (3,1) \}

We are interested in two subgroups H_1 and H_2, both isomorphic to cyclic group:Z4, that are automorphic subgroups (i.e., H_1 can be sent to H_2 by an automorphism of G). We have:

H_1 = \{ (0,0), (1,0), (2,0), (3,0) \}

H_2 = \{ (0,0), (1,1), (2,0), (3,1) \}

The corresponding quotient groups in both cases are isomorphic to cyclic group:Z2.

Note that both of these are direct factors of the whole group. They are precisely the non-characteristic subgroups of the whole group of order four. There is also a characteristic subgroup of order four, given by \{ (0,0), (0,1), (2,0), (2,1) \}, which is described on the page first omega subgroup of direct product of Z4 and Z2.

Cosets

H_1 is a normal subgroup of G, so its left cosets coincide with its right cosets. The two cosets are as follows:

\! \{ (0,0), (1,0), (2,0), (3,0) \}, \{ (0,1), (1,1), (2,1), (3,1) \}

H_2 is a normal subgroup of G, so its left cosets coincide with its right cosets. The two cosets are as follows:

\! \{ (0,0), (1,1), (2,0), (3,1) \}, \{ (0,1), (1,0), (2,1), (3,0) \}

Arithmetic functions

Function Value Explanation
order of whole group 8
order of subgroup 4
index of subgroup 2
size of conjugacy class of subgroup 1
number of conjugacy classes in automorphism class of subgroup 2
size of automorphism class of subgroup 2

Dual subgroup

We know that subgroup lattice and quotient lattice of finite abelian group are isomorphic, which means that there must exist an automorphism class of subgroups of G that plays the role of dual subgroups to H_1 and H_2 -- in particular, that is isomorphic to the quotient group G/H_1 and its quotient group is isomorphic to H_1. This automorphism class of subgroups is given by non-characteristic order two subgroups of direct product of Z4 and Z2.

Subgroup properties

Invariance under automorphisms and endomorphisms: basic properties

Property Meaning Satisfied? Explanation GAP verification (set G := DirectProduct(CyclicGroup(4),CyclicGroup(2)); H := Image(Embedding(G,1));)
normal subgroup invariant under all inner automorphisms Yes IsNormal(G,H); using IsNormal
characteristic subgroup invariant under all automorphisms No IsCharacteristicSubgroup(G,H); using IsCharacteristicSubgroup
fully invariant subgroup invariant under all endomorphisms No IsFullinvariant(G,H); using IsFullinvariant

Factorization-related properties

Property Meaning Satisfied? Explanation
direct factor one of the factors in an internal direct product Yes The complement is \{ (0,0), (0,1) \}
retract has a normal complement Yes Follows from being a direct factor
complemented normal subgroup has a normal complement Yes Follows from being a direct factor
permutably complemented subgroup has a normal complement Yes Follows from being a direct factor
AEP-subgroup any automorphism of the subgroup extends to an automorphism of the whole group Yes direct factor implies AEP

GAP implementation

The group-subgroup pair can be defined using the DirectProduct, CyclicGroup, Image, and Embedding as:

G := DirectProduct(CyclicGroup(4),CyclicGroup(2)); H := Image(Embedding(G,1));