Klein four-subgroup of alternating group:A5
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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) alternating group:A5 (see subgroup structure of alternating group:A5).
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Let be the alternating group:A5, i.e., the alternating group (the group of even permutations) on the set
.
has order
.
Consider the subgroup:
This subgroup is isomorphic to the Klein four-group. There are five conjugates (including the subgroup itself) depending on which of the five points is fixed:
Fixed point | Subgroup name here | Elements of subgroup |
---|---|---|
1 | ![]() |
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2 | ![]() |
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3 | ![]() |
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4 | ![]() |
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5 | ![]() |
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Contents
Arithmetic functions
Function | Value | Explanation |
---|---|---|
order of the whole group | 60 | The order is ![]() |
order of the subgroup | 4 | |
index of the subgroup | 15 | |
size of conjugacy class of subgroups (equal to index of normalizer) | 3 | |
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 | ![]() |
A4 in A5 | alternating group:A4 |
centralizer | the subgroup itself | current page | Klein four-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 subgroup for the prime ![]() |
A ![]() ![]() ![]() ![]() |
The order of this subgroup is 4, which is the largest power of 2 dividing the order of the group. |
Related subgroups
Intermediate subgroups
Value of intermediate subgroup (descriptive) | Isomorphism class of intermediate subgroup | Number of conjugacy classes of intermediate subgroup fixing subgroup and whole group | Subgroup in intermediate subgroup | Intermediate subgroup in whole group |
---|---|---|---|---|
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alternating group:A4 | 1 | V4 in A4 | A4 in A5 |
Smaller subgroups
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 |
---|---|---|---|---|
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cyclic group:Z2 | 1 | Z2 in V4 | subgroup generated by double transposition in A5 |
Subgroup properties
Sylow and corollaries
The subgroup is a 2-Sylow subgroup, so many properties follow as a corollary of that.
Property | Meaning | Why being Sylow implies the property |
---|---|---|
order-conjugate subgroup | conjugate to any subgroup of the same order | Sylow implies order-conjugate |
isomorph-conjugate subgroup | conjugate to any subgroup isomorphic to it | (via order-conjugate) |
automorph-conjugate subgroup | conjugate to any subgroup automorphic to it | (via isomorph-conjugate) |
intermediately isomorph-conjugate subgroup | conjugate to any subgroup isomorphic to it inside any intermediate subgroup | Sylow implies intermediately isomorph-conjugate |
intermediately automorph-conjugate subgroup | automorph-conjugate in every intermediate subgroup | (via intermediately isomorph-conjugate) |
pronormal subgroup | conjugate to any conjugate subgroup in their join | Sylow implies pronormal |
weakly pronormal subgroup | (via pronormal) | |
paranormal subgroup | (via pronormal) | |
polynormal subgroup | (via pronormal) |
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 | ||
self-normalizing subgroup | equals its normalizer in the whole group | No | normalizer is A4 in A5 | |
self-centralizing subgroup | contains its centralizer in the whole group | Yes |