# Klein four-subgroup of M16

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) Klein four-group and the group is (up to isomorphism) M16 (see subgroup structure of M16).
The subgroup is a normal subgroup and the quotient group is isomorphic to cyclic group:Z4.
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## Definition

We consider the group:

$G = M_{16} = \langle a,x \mid a^8 = x^2 = e, xax = a^5 \rangle$

with $e$ denoting the identity element.

This is a group of order 16, with elements:

$\! \{ e, a, a^2, a^3, a^4, a^5, a^6, a^7, x, ax, a^2x, a^3x, a^4x, a^5x, a^6x, a^7x \}$

We are interested in the subgroup:

$\! H = \{ e, x, a^4, a^4x \}$

This is a subgroup of order four isomorphic to the Klein four-group, i.e., it is an elementary abelian group of prime-square order for the prime 2: all its non-identity elements have order 2.

## Cosets

The subgroup has order 4 and index 4, so it has four cosets. Since it is a normal subgroup, the left cosets coincide with the right cosets:

$\! \{ e, x, a^4, a^4x \}, \{ a, ax, a^5, a^5x \}, \{ a^2, a^2x, a^6, a^6x \}, \{ a^3, a^3x, a^7, a^7x \}$

## 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 4
index 4
size of conjugacy class 1
number of conjugacy classes in automorphism class 1

## Effect of subgroup operators

Function Value as subgroup (descriptive) Value as subgroup (link) Value as group
normalizer whole group $\langle a,x \rangle$ M16
centralizer $\langle a^2,x \rangle$ direct product of Z4 and Z2 in M16 direct product of Z4 and Z2
normal core the subgroup itself current page Klein four-group
normal closure the subgroup itself current page Klein four-group
characteristic core the subgroup itself current page Klein four-group
characteristic closure the subgroup itself current page Klein four-group
commutator with whole group $\langle a^4 \rangle$ derived subgroup of M16 cyclic group:Z2

## Subgroup-defining functions

The subgroup is a characteristic subgroup of the whole group and arises as a result of many subgroup-defining functions on the whole group. Some of these are given below.

Subgroup-defining function Meaning in general Why it takes this value
first omega subgroup subgroup generated by the elements of order $p$, where $p$ is the underlying prime (in this case $p = 2$) In fact, the elements $a^4, x, a^4x$ are the only elements of order two in M16, and they form a subgroup along with the identity element.
join of elementary abelian subgroups of maximum order subgroup generated by all the elementary abelian subgroups of maximum order In this case, it is the unique elementary abelian subgroup of maximum order.

## Subgroup properties

### Invariance under automorphisms and endomorphisms

Property Meaning Satisfied? Explanation
normal subgroup invariant under inner automorphisms Yes Precisely the set of elements of order two.
characteristic subgroup invariant under all automorphisms Yes Precisely the set of elements of order two.
fully invariant subgroup invariant under all endomorphisms Yes Precisely the set of elements of order two.
image-closed characteristic subgroup image under any surjective homomorphism from whole group is characteristic in target group No Taking quotient by derived subgroup of M16, we get a direct factor of direct product of Z4 and Z2, which is hence not characteristic in it.
image-closed fully invariant subgroup image under any surjective homomorphism from whole group is fully invariant in target group No Follows from not being image-closed characteristic.
verbal subgroup generated by set of words No Follows from its not being an image-closed characteristic subgroup
isomorph-free subgroup no other isomorphic subgroup Yes It is precisely the subgroup of elements of order at most two.
isomorph-normal subgroup Every isomorphic subgroup is normal Yes Follows from being isomorph-free.
homomorph-containing subgroup contains all homomorphic images Yes Any homomorphic image must comprise elements of order one or two, all of which are in this subgroup.
1-endomorphism-invariant subgroup invariant under all 1-endomorphisms of the group Yes Under any 1-endomorphism, all elements must go to elements of order 1 or 2.
1-automorphism-invariant subgroup invariant under all 1-automorphisms of the group Yes Follows from being 1-endomorphism-invariant.
quasiautomorphism-invariant subgroup invariant under all quasiautomorphisms Yes Follows from being 1-automorphism-invariant