A3 in S5
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:Z3 and the group is (up to isomorphism) symmetric group:S5 (see subgroup structure of symmetric group:S5).
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Consider the subgroup:
has a total of 10 conjugate subgroups (including itself) and the subgroups are parametrized by subsets of size 3 in describing the support of the 3-cycles. The complementary subset of size two is fixed point-wise by that conjugate subgroup:
|Subset of size 3||Complementary subset of size 2 fixed point-wise by the subgroup||Conjugate of|
Each of these subgroups is isomorphic to cyclic group:Z3.
|order of the whole group||120||. See symmetric group:S5.|
|[[order of a group|order of the subgroup||3|
|index of the subgroup||40||Follows from Lagrange's theorem|
|size of conjugacy class of subgroups = index of normalizer||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||direct product of S3 and S2 in S5||dihedral group:D12|
|centralizer||Z6 in S5||cyclic group:Z6|
|normal core||trivial subgroup||--||trivial group|
|normal closure||the alternating subgroup||A5 in S5||alternating group:A5|
|characteristic core||trivial subgroup||--||trivial group|
|characteristic closure||the alternating subgroup||A5 in S5||alternating group:A5|
Conjugacy class-defining functions
|Conjugacy class-defining function||What it means in general||Why it takes this value|
|Sylow subgroup for the prime||A -Sylow subgroup is a subgroup whose order is a power of and index is relatively prime to . Sylow subgroups exist and Sylow implies order-conjugate, i.e., all -Sylow subgroups are conjugate to each other.||The order of this subgroup is 3, which is the largest power of 3 dividing the order of the group.|