Z2 in V4

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) Klein four-group (see subgroup structure of Klein four-group).
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

We are interested in the Klein four-group:

$G := \{ e,a,b,c \}$

where $e$ is the identity element and $a,b,c$ are all non-identity elements of order 2.

We are interested in the subgroup:

$H = H_0 = \{ e, a\}$

as well as the other two subgroups automorphic to $H$:

$H_1 = \{ e,b \}, H_2 = \{ e, c \}$

Direct product and direct factor interpretation

This article describes the subgroup $H$ in the group $G$, where $G$ is the Klein four-group:

$G := \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$

Denoted additively, the four elements of $G$ are:

$\! \{ (0,0), (0,1), (1,0), (1,1) \}$

where the addition is modulo 2 in each coordinate separately.

and $H = H_0$ is the two-element subgroup:

$\! \{ (0,0), (1,0) \}$

$H$ is a normal subgroup (since $G$ is an abelian group), so has no other conjugate subgroups. There are two other automorphic subgroups to $H$ in $G$:

$\! H_1 = \{ (0,0), (0,1) \}, H_2 = \{ (0,0), (1,1) \}$

The quotient group $G/H$ is isomorphic to cyclic group:Z2.

Cosets

Each of $H$, $H_1$ and $H_2$ is a normal subgroup and each has two cosets: the subgroup itself and the rest of the group. There is a total of six cosets. In fact, every subset of size two in the whole group (total of $\binom{4}{2} = 6$ such subsets) is a coset of exactly one of these three subgroups.

Complements

Any two of the three subgroups $H, H_1, H_2$ are permutable complements of each other. Since they're all normal, this means that $G$ is an internal direct product of any two of them.

Properties related to complementation

Property	Meaning	Satisfied?	Explanation	Comment
complemented normal subgroup	normal subgroup with permutable complement	Yes
complemented characteristic subgroup	characteristic subgroup with permutable complement	No	not a characteristic subgroup
direct factor	normal subgroup with normal complement	Yes
retract	subgroup with normal complement	Yes
permutably complemented subgroup	subgroup with permutable complement	Yes
lattice-complemented subgroup	subgroup with lattice complement	Yes

Arithmetic functions

Function	Value	Explanation
order of whole group	4
order of subgroup	2
index	2
size of conjugacy class	1
number of conjugacy classes in automorphism class	3
size of automorphism class	3

Effect of subgroup operators

Function	Value as subgroup (descriptive)	Value as subgroup (link)	Value as group
normalizer	the whole group	--	Klein four-group
centralizer	the whole group	--	Klein four-group
normal core	the subgroup itself	current page	cyclic group:Z2
normal closure	the subgroup itself	current page	cyclic group:Z2
characteristic core	trivial subgroup	--	trivial group
characteristic closure	the whole group	--	Klein four-group
commutator with whole group	trivial subgroup	--	trivial group

Subgroup properties

Invariance under automorphisms and endomorphisms: properties

Property	Meaning	Satisfied?	Explanation	Comment
normal subgroup	invariant under all inner automorphisms, equals its conjugate subgroups	Yes	abelian implies every subgroup is normal
characteristic subgroup	invariant under all automorphisms	No	Not invariant under automorphism that interchanges coordinates, i.e., that sends $(a,b)$ to $(b,a)$	See also normal not implies characteristic. This is the smallest example of a normal subgroup that is not characteristic and is part of a general family of such examples.
coprime automorphism-invariant subgroup	invariant under automorphisms of order coprime to that of the group	No	The group has an automorphism of order three that cycles the three non-identity elements -- this cyclically permutes the three automorphs of $H$
cofactorial automorphism-invariant subgroup	invariant under all automorphisms whose order has no prime factors other than those of the whole group	No	The coordinate interchange automorphism does not preserve the subgroup, and has order 2
subgroup-cofactorial automorphism-invariant subgroup	invariant under all automorphisms whose order has no prime factors other than those of the subgroup	No	The coordinate interchange automorphism does not preserve the subgroup, and has order 2
retraction-invariant subgroup	invariant under all retractions, i.e., endomorphisms that equal their own square	No	Consider the retraction with kernel $H_1$ and image $H_2$ (realized as projection onto $H_2$ in the realization of $G$ as an internal direct product of $H_1$ and $H_2$). Under this, $H$ maps to $H_2$.

Resemblance-based properties

Property	Meaning	Satisfied?	Explanation	Comment
order-isomorphic subgroup	isomorphic to any subgroup of the same order	Yes	follows since order is prime, and there is only one isomorphism class of a group of prime order
order-automorphic subgroup	any subgroup of the same order is automorphic to it	Yes	the three subgroups of order 2 are precisely $H, H_1, H_2$
isomorph-automorphic subgroup	any isomorphic subgroup is automorphic	Yes	the three subgroups of order 2 are precisely $H, H_1, H_2$
isomorph-conjugate subgroup	conjugate to every isomorphic subgroup	No	distinct normal subgroups -- cannot be conjugate
order-conjugate subgroup	conjugate to every subgroup of the same order	No	distinct normal subgroups -- cannot be conjugate
automorph-conjugate subgroup	conjugate to every automorphic subgroup	No	distinct normal subgroups -- cannot be conjugate