Twisted S3 in A5
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) symmetric group:S3 and the group is (up to isomorphism) alternating group:A5 (see subgroup structure of alternating group:A5).
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Contents
Definition
Let be the alternating group:A5, i.e., the alternating group (the group of even permutations) on the set
.
has order
.
Consider the subgroup:
One way of thinking of is as follows: we first take the symmetric group on the set
, which is isomorphic to symmetric group:S3, and is a subgroup of symmetric group:S5 but not of alternating group:a5. We then define a homomorphism
from this to the subgroup
of symmetric group:S5 as follows: every even permutation goes to the identity, and every odd permutation goes to
. We then consider the image of the group under the map
. The upshot: even permutations remain as they are, and odd permutations get multiplied by
.
We see by construction that the map has trivial kernel, so
is isomorphic to symmetric group:S3. It is also clear that since
and
have the same sign, the product permutation is even and hence in the alternating group.
has a conjugate corresponding to every possible partition of
into a subset of size 3 and a subset of size 2:
Subset of size 3 | Subset of size 2 | Corresponding conjugate of ![]() |
---|---|---|
1,2,3 | 4,5 | ![]() |
1,2,4 | 3,5 | ![]() |
1,2,5 | 3,4 | ![]() |
1,3,4 | 2,5 | ![]() |
1,3,5 | 2,4 | ![]() |
1,4,5 | 2,3 | ![]() |
2,3,4 | 1,5 | ![]() |
2,3,5 | 1,4 | ![]() |
2,4,5 | 1,3 | ![]() |
3,4,5 | 1,2 | ![]() |
Arithmetic functions
Function | Value | Explanation |
---|---|---|
order of whole group | 60 | See alternating group:A5 |
order of subgroup | 6 | See symmetric group:S3 |
index of the subgroup | 10 | |
size of conjugacy class | 10 | |
number of conjugacy classes in automorphism class | 1 |
Effect of subgroup operators
In the table below, we provide values specific to .
Function | Value as subgroup (descriptive) | Value as subgroup (link) | Value as group |
---|---|---|---|
normalizer | the subgroup itself | current page | symmetric group:S3 |
centralizer | trivial subgroup | -- | trivial group |
normal core | trivial subgroup | -- | trivial group |
normal closure | the whole group | -- | alternating group:A5 |
characteristic core | trivial subgroup | -- | trivial group |
characteristic closure | the whole group | -- | alternating group:A5 |
Conjugacy class-defining functions
Conjugacy class-defining function | What it means in general | Why it takes this value |
---|---|---|
Sylow normalizer for the prime ![]() |
Normalizer of a ![]() ![]() |
A 3-Sylow subgroup is A3 in A5, given by ![]() |
Related subgroups
Intermediate subgroups
There are no intermediate subgroups, because this subgroup is a maximal subgroup.
Smaller subgroups
The values given below are specific to .
Value of smaller subgroup (descriptive) | Isomorphism class of smaller subgroup | Number of conjugacy classes of smaller subgroup fixing subgroup and whole group | Smaller subgroup in subgroup | Smaller subgroup in whole group |
---|---|---|---|---|
![]() |
cyclic group:Z3 | 1 | A3 in S3 | A3 in A5 |
![]() |
cyclic group:Z2 | 3 | S2 in S3 | subgroup generated by double transposition in A5 |
Description in terms of alternative interpretations of the whole group
Description of ![]() |
Corresponding description of ![]() |
---|---|
special linear group of degree two over field:F4 | Group of monomial matrices of determinant 1, i.e., matrices with determinant 1 and where each row has exactly one nonzero entry and each column has exactly one nonzero entry. Alternatively, one of the conjugate subgroups can be thought of as a copy of ![]() ![]() |
projective special linear group of degree two over field:F5 | If we think of the field ![]() ![]() ![]() ![]() ![]() |
Subgroup properties
Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|
normal subgroup | equals all its conjugate subgroups | No | (see above for other conjugate subgroups) | |
2-subnormal subgroup | normal subgroup in its normal closure | No | ||
subnormal subgroup | series from subgroup to whole group, each normal in next | No | ||
contranormal subgroup | normal closure is whole group | Yes | ||
abnormal subgroup | For any element ![]() ![]() |
Yes | Follows from being a Sylow normalizer, see Sylow normalizer implies abnormal | |
weakly abnormal subgroup | Yes | Follows from being abnormal | ||
self-normalizing subgroup | equals normalizer in the whole group | Yes | ||
self-centralizing subgroup | contains its centralizer in the whole group | Yes | ||
subgroup whose join with any distinct conjugate is the whole group | join of the subgroup with any distinct conjugate subgroup is the whole group | Yes | ||
maximal subgroup | no proper subgroup containing it | Yes | Note that in a finite solvable group, any maximal subgroup has prime power index (see maximal subgroup has prime power index in finite solvable group). The fact that we have a maximal subgroup here whose index is not a prime power is consistent with the fact that alternating group:A5 is not solvable. | |
pronormal subgroup | any conjugate to it is conjugate in their join | Yes | ||
weakly pronormal subgroup | Yes | |||
paranormal subgroup | Yes | |||
polynormal subgroup | Yes |
Resemblance-based properties
Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|
order-conjugate subgroup | conjugate to any subgroup of the whole group of the same order | Yes | See subgroup structure of alternating group:A5 | |
isomorph-conjugate subgroup | conjugate to every subgroup of the whole group isomorphic to it | Yes | See subgroup structure of alternating group:A5 | |
automorph-conjugate subgroup | conjugate to every subgroup of the whole group automorphic to it | Yes | Follows from being a Sylow normalizer | Sylow implies automorph-conjugate |