Derived 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) cyclic group: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 direct product of Z4 and Z2.
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Contents
Definition
We consider the group:
with denoting the identity element.
This is a group of order 16, with elements:
We are interested in the subgroup:
This is a cyclic subgroup of order two, i.e., it is isomorphic to cyclic group:Z2. The quotient group is isomorphic to the abelian group direct product of Z4 and Z2.
Cosets
The subgroup is a normal subgroup and hence its left cosets coincide with its right cosets. The eight cosets are given below:
Complements
The subgroup has no permutable complements. Since it is a normal subgroup, this also means it has no lattice complements.
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 | 2 | |
index | 8 | |
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 | M16 | |
centralizer | whole group | M16 | |
normal core | the subgroup itself | current page | cyclic group:Z2 |
normal closure | the subgroup itself | current page | cyclic group:Z2 |
characteristic core | the subgroup itself | current page | cyclic group:Z2 |
characteristic closure | the subgroup itself | current page | cyclic group:Z2 |
commutator with whole group | trivial subgroup | -- | trivial group |
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 |
---|---|---|
derived subgroup | subgroup generated by all commutators between elements of the group | The only commutators are the identity element and . |
socle | subgroup generated by the minimal normal subgroups. For a group of prime power order , this is the same as -- the identity element and all elements of prime order in the center | It is the unique minimal normal subgroup. The center is in fact , and this is precisely the set of elements of order at most 2 in the center. |
second agemo subgroup | subgroup generated by all powers where is the underlying prime (here ) | The fourth powers are precisely the identity and . |
Subgroup properties
Invariance under automorphisms and endomorphisms
Property | Meaning | Satisfied? | Explanation |
---|---|---|---|
normal subgroup | invariant under inner automorphisms | Yes | Follows from its being the derived subgroup. |
characteristic subgroup | invariant under all automorphisms | Yes | Follows from its being the derived subgroup. |
fully invariant subgroup | invariant under all endomorphisms | Yes | Follows from its being the derived subgroup. |
verbal subgroup | generated by set of words | Yes | Follows from its being the derived subgroup. |
image-closed characteristic subgroup | image under any surjective homomorphism from whole group is characteristic in target group | Yes | Follows from its being a verbal subgroup. |
image-closed fully invariant subgroup | image under any surjective homomorphism from whole group is fully invariant in target group | Yes | Follows from its being a verbal subgroup. |
isomorph-free subgroup | no other isomorphic subgroup | No | is another isomorphic subgroup. |
isomorph-normal subgroup | every isomorphic subgroup is normal | No | is a non-normal isomorphic subgroup. |
normal-isomorph-free subgroup | no other isomorphic normal subgroup | Yes | Follows from its being the socle. |
homomorph-containing subgroup | contains all homomorphic images | No | Follows from not being isomorph-free. |
1-endomorphism-invariant subgroup | invariant under all 1-endomorphisms of the group | Yes | Follows from its being precisely the set of fourth powers. |
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 |
GAP implementation
The group and subgroup can be constructed using GAP's SmallGroup and DerivedSubgroup functions as follows:
G := SmallGroup(16,6); H := DerivedSubgroup(G);
The GAP display looks as follows:
gap> G := SmallGroup(16,6); H := DerivedSubgroup(G); <pc group of size 16 with 4 generators> Group([ f4 ])Here is some GAP implementation to verify assertions made on this page: [SHOW MORE]