Abelian minimal normal subgroup and core-free maximal subgroup are permutable complements
- and intersect trivially
- The induced action of on by left multiplication, is equivalent to the regular action of a group on itself.
Further information: Core-free subgroup
A subgroup of a group is termed core-free if its normal core in the whole group is trivial, or equivalently, if it does not contain any nontrivial normal subgroup of the whole group.
Further information: Maximal subgroup
A maximal subgroup of a group is a proper subgroup that is not contained in any bigger proper subgroup.
Further information: Primitive group A primitive group is a group that possesses a core-free maximal subgroup.
- Abelian normal subgroup and core-free subgroup generate whole group implies they intersect trivially
Further information: plinth theorem
The plinth theorem is a similar result to the case of two distinct minimal nomral subgroups.
More results under similar hypotheses
Proceeding further along the lines of this proof, one can show that:
- Abelian permutable complement to core-free subgroup is self-centralizing
- Primitive implies Fitting-free or elementary Abelian Fitting subgroup: In other words, is maximal among Abelian normal subgroups, and also is the only nontrivial Abelian normal subgroup, and is the only nontrivial nilpotent normal subgroup.
Breakdown of similar results
Given: A primitive group with core-free maximal subgroup , and an Abelian minimal normal subgroup
To prove: and are permutable complements: and is trivial
Proof: Since is normal, is a subgroup of containing . Since is maximal, either or . If , then , but this cannot happen because does not contain any nontrivial normal subgroup of . Hence .
Thus is an Abelian normal subgroup and is a core-free subgroup that together generate . Applying the fact (1) stated above, is trivial, so and are permutable complements.