Normal Klein four-subgroup of symmetric group:S4
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) symmetric group:S4 (see subgroup structure of symmetric group:S4).
The subgroup is a normal subgroup and the quotient group is isomorphic to symmetric group:S3.
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This article discusses the normal subgroup in the symmetric group of degree four comrpising the identity and the three double transpositions.
We let be the symmetric group of degree four, acting on and be the subgroup of given by:
The six cosets of this subgroup are as follows:
COMPLEMENTS TO NORMAL SUBGROUP: TERMS/FACTS TO CHECK AGAINST:
TERMS: permutable complements | permutably complemented subgroup | lattice-complemented subgroup | complemented normal subgroup (normal subgroup that has permutable complement, equivalently, that has lattice complement) | retract (subgroup having a normal complement)
FACTS: complement to normal subgroup is isomorphic to quotient | complements to abelian normal subgroup are automorphic | complements to normal subgroup need not be automorphic | Schur-Zassenhaus theorem (two parts: normal Hall implies permutably complemented and Hall retract implies order-conjugate)
Each of these is isomorphic to the quotient group , since complement to normal subgroup is isomorphic to quotient.
|complemented normal subgroup||normal subgroup, has a permutable complement||Yes||See above list of complements|
|permutably complemented subgroup||has a permutable complement||Yes|
|lattice-complemented subgroup||has a lattice complement||Yes|
|retract||has a normal complement||No|
|direct factor||normal subgroup with normal complement||No|
|complemented characteristic subgroup||characteristic subgroup with permutable complement||Yes|
|complemented fully invariant subgroup||fully invariant subgroup with permutable complement||Yes|
|order of whole group||24|
|order of subgroup||4|
|size of conjugacy class||1|
|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||whole group||--||symmetric group:S4|
|centralizer||the subgroup itself||current page||Klein four-group|
|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|
The subgroup is a characteristic subgroup of the whole group and arises as the result of many common subgroup-defining functions on the whole group.
|Subgroup-defining function||What it means in general||Why it takes this value|
|second derived subgroup||second member of derived series. In other words, derived subgroup of derived subgroup, i.e.,||The derived subgroup is A4 in S4, and the derived subgroup of the derived subgroup is the Klein four-subgroup. The corresponding quotient groups are cyclic group:Z2 and cyclic group:Z3.|
|2-Sylow core or 2-core||largest normal subgroup whose order is a power of 2. Equivalently, normal core of any 2-Sylow subgroup||There are three 2-Sylow subgroups, each isomorphic to dihedral group:D8, and their intersection is this subgroup.|
|socle||join of all minimal normal subgroups||In fact, it is the unique minimal normal subgroup (hence the monolith) -- the whole group is a monolithic group.|
Description in terms of alternative interpretations of the whole group
|Interpretation of||Corresponding interpretation of|
|As symmetric group of degree four||Subset comprising identity and double transpositions|
|As projective general linear group of degree two over field:F3||images of semisimple elements of determinant 1 (note: this does not usually form a subgroup, but does for this field and this degree)|
|As group of all isometries of the regular tetrahedron (including orientation-reversing ones)||identity, and isometries obtained as follows: consider two opposite edges (i.e., edges that share no vertex). Take the line joining their midpoints. Now, do a half turn (rotation by 180 degrees) about this line.|
|As group of all orientation-preserving isometries of the cube, or the regular octahedron||identity, and rotations by 180 degrees about the three axes of the cube.|
|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|
|dihedral group:D8||3||Klein four-subgroups of dihedral group:D8||D8 in S4|
|alternating group:A4||1||Klein four-subgroup of alternating group:A4||A4 in S4|
|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:Z2||3||Z2 in V4||subgroup generated by double transposition in S4|
Invariance under automorphisms and endomorphisms: properties
|normal subgroup||invariant under all inner automorphisms, equals all conjugate subgroups||Yes||identity element plus conjugacy class of all double transpositions|
|characteristic subgroup||invariant under all automorphisms||Yes||Since the whole group is a complete group, all automorphisms are inner|
|fully invariant subgroup||invariant under all endomorphisms||Yes||Since it is the second derived subgroup; also because monolith is fully invariant in co-Hopfian|
|verbal subgroup||set of all occurrences of a bunch of words||Yes||Since it is the second derived subgroup|
|image-closed fully invariant subgroup||under any homomorphic image of whole group, image is fully invariant||Yes||verbal implies image-closed fully invariant|
|image-closed characteristic subgroup||under any homomorphic image of whole group, image is characteristic||Yes||(Via image-closed fully invariant)|
|homomorph-containing subgroup||contains any homomorphic image of it in the whole group||No||There are cyclic subgroups of order two in the whole group not in the subgroup, namely, those generated by single transpositions. These cyclic subgroups are homomorphic images of .|
|isomorph-containing subgroup, isomorph-free subgroup||contains/equals every isomorphic subgroup (properties equivalent for finite groups)||No||The subgroup and its conjugates, non-normal Klein four-subgroups of symmetric group:S4, are isomorphic to it.|
|intermediately characteristic subgroup||characteristic in every intermediate subgroup||No||Not characteristic in , where it sits in like Klein four-subgroups of dihedral group:D8|
Interpretations from subgroup upwards
Interpretation in terms of Cayley's theorem
We can think of the embedding of in in terms of Cayley's theorem. Specifically, think of starting with as an abstract Klein four-group whose elements are labeled . Left multiplication by elements of induces precisely the identity and the three double transpositions on as a set. This thus makes a subgroup of the symmetric group on , which is .
Note that the other, non-normal Klein four-subgroup cannot be interpreted this way because its non-identity elements are not fixed-point-free. However, the cyclic four-subgroup of the symmetric group of degree four can be embedded in this way.
Interpretation in terms of holomorph
We can think of the embedding of in as the abstract group sitting inside its holomorph. This is because is the semidirect product of and its automorphism group, which is , which is isomorphic to the symmetric group of degree three.