Non-characteristic order two subgroups of direct product of Z4 and Z2
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) 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:Z4.
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Contents
Definition
The group 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:
We are interested in two subgroups and , both isomorphic to cyclic group:Z2, that are automorphic subgroups (i.e., can be sent to by an automorphism of ). We have:
The corresponding quotient groups in both cases are isomorphic to cyclic group:Z4.
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 two. There is also a characteristic subgroup of order two, given by , which is described on the page first agemo subgroup of direct product of Z4 and Z2.
Cosets
is a normal subgroup of , so its left cosets coincide with its right cosets. The four cosets are as follows:
is a normal subgroup of , so its left cosets coincide with its right cosets. The four cosets are as follows:
Arithmetic functions
Function | Value | Explanation |
---|---|---|
order of whole group | 8 | |
order of subgroup | 2 | |
index of subgroup | 4 | |
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 that plays the role of dual subgroups to and -- in particular, that is isomorphic to the quotient group and its quotient group is isomorphic to . This automorphism class of subgroups is given by Z4 in 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,2));) |
---|---|---|---|---|
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 |
Property | Meaning | Satisfied? | Explanation |
---|---|---|---|
direct factor | one of the factors in an internal direct product | Yes | The complement is |
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,2));