Permutable implies subnormal in finite

Statement
Any permutable subgroup of a finite group is a subnormal subgroup. In particular, it is a permutable subnormal subgroup.

Similar facts

 * Conjugate-permutable implies subnormal in finite
 * Permutable implies ascendant
 * Permutable implies locally subnormal
 * Permutable implies subnormal in finitely generated

Generalizations of proof technique

 * Max-equivalent to normal and intermediate subgroup condition implies subnormal in finite: Other examples of this proof technique are:
 * Conjugate-permutable implies subnormal in finite

Facts used

 * 1) uses::Maximal permutable implies normal
 * 2) uses::Permutability satisfies intermediate subgroup condition

Proof
Given: A finite group $$G$$, a permutable subgroup $$H$$ of $$G$$.

To prove: $$H$$ is subnormal in $$G$$.

Proof: We prove this by induction on the order of $$G$$.

If $$H = G$$, we are done. Otherwise:


 * 1) Let $$K$$ be a maximal element among the proper permutable subgroups of $$G$$ containing $$H$$. By fact (1), $$K$$ is normal in $$G$$.
 * 2) By fact (2), $$H$$ is permutable in $$K$$, so induction on the order yields that $$H$$ is subnormal in $$K$$.
 * 3) Thus, $$H$$ is subnormal in $$K$$ which is normal in $$G$$, so $$H$$ is subnormal in $$G$$.