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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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 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:
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 | 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 | M16 | |
centralizer | 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 | 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 , where is the underlying prime (in this case ) | In fact, the elements 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 |