Maximal conjugate-permutable implies normal
This article gives the proof of a maximality equivalence. In other words, there are two subgroup properties: a stronger one (Normal subgroup (?)) and a weaker one (Conjugate-permutable subgroup (?)). However, any subgroup maximal among the proper subgroups with the weaker property also has the stronger property, and is thus also maximal among proper subgroups with the stronger property.
View other such statements
- Pronormal implies self-conjugate-permutable
- Maximal implies self-conjugate-permutable
- Conjugate-permutable and self-conjugate-permutable implies normal
- Conjugate-permutability is conjugate-join-closed: If is a conjugate-permutable subgroup of , and , then is also a conjugate-permutable subgroup of .
- Product of conjugates is proper: If is a proper subgroup of , and , then is a proper subset of .
Given: A group , a proper conjugate-permutable subgroup of such that is not contained in any proper conjugate-permutable subgroup of .
To prove: is normal in : for any , .
- (Fact used: fact (1), conjugate-permutability is conjugate-join-closed): Suppose . Then, since is conjugate-permutable in , fact (1) tells us that is also conjugate-permutable in .
- (Given data used: is maximal conjugate-permutable): By our assumption, either or .
- (Fact used: fact (2), product of conjugates is proper): If , we have , which yields a contradiction by fact (2).
- Combining steps (2) and (3), we see that , forcing , as required.