Direct product of Z4 and Z2 in M16

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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) direct product of Z4 and Z2 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:Z2.
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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, a^2, a^4, a^6, x, a^2x, a^4x, a^6x \} = \langle a^2,x \rangle

This is a subgroup of order eight isomorphic to direct product of Z4 and Z2, where the cyclic four-subgroup is \langle a^2 \rangle and the cyclic two-subgroup is \langle x \rangle.

Cosets

The subgroup is a subgroup of index two, hence it is a normal subgroup (see index two implies normal) and in particular its left cosets coincide with its right cosets. The two cosets are:

H =\{ e, a^2, a^4, a^6, x, a^2x, a^4x, a^6x \}, G \setminus H = \{ a, a^3, a^5, a^7, ax, a^3x, a^5x, 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 8
index 2
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 the subgroup itself current page direct product of Z4 and Z2
normal core the subgroup itself current page direct product of Z4 and Z2
normal closure the subgroup itself current page direct product of Z4 and Z2
characteristic core the subgroup itself current page direct product of Z4 and Z2
characteristic closure the subgroup itself current page direct product of Z4 and Z2
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
second omega subgroup For a group of prime power order, subgroup generated by all elements whose order divides the square of the prime All elements in the subgroup have order at most 4, all elements outside it have order 8.
join of abelian subgroups of maximum rank For a group of prime power order, this is the subgroup generated by all the abelian subgroups of maximum rank. The maximum possible rank of an abelian subgroup is 2. There are two such subgroups, V4 in M16 and this subgroup, and this is the bigger of the two.

Subgroup properties

Invariance under automorphisms and endomorphisms

Property Meaning Satisfied? Explanation
normal subgroup invariant under inner automorphisms Yes Index two implies normal
characteristic subgroup invariant under all automorphisms Yes Follows from being the largest abelian subgroup of rank two.
fully invariant subgroup invariant under all endomorphisms Yes Follows from its being the second omega subgroup -- all elements in the subgroup have order at most four, all elements outside it have order bigger than four.
image-closed characteristic subgroup image under any surjective homomorphism from whole group is characteristic in target group No Taking quotient by center of M16, we get Z2 in V4, which is not characteristic.
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 set of elements of order at most four.
isomorph-normal subgroup all isomorphic subgroups are normal in the whole group Yes Follows from being isomorph-free
homomorph-containing subgroup contains all homomorphic images Yes Any homomorphic image must comprise elements of order at most four, 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 at most 4.
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