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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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|
|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 -Sylow subgroup. Since Sylow implies order-conjugate (any two -Sylow subgroups are conjugate), any two normalizers thereof are also conjugate.||A 3-Sylow subgroup is A3 in A5, given by , and its normalizer is precisely this subgroup.|
There are no intermediate subgroups, because this subgroup is a maximal subgroup.
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 in , via the embedding of field:F2 in field:F4.
|projective special linear group of degree two over field:F5||If we think of the field as a two-dimensional vector space over field:F5, its multiplicative group embeds in . The intersection with corresponds to the elements whose sixth power is the identity. Projecting down to gives a cyclic group of order three, which is precisely the subgroup A3 in A5. There is an automorphism arising from the Frobenius upstairs, and introducing this as an element of gives the whole group.|
|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 , we have||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|
|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|