Permutable not implies normal
This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., permutable subgroup) need not satisfy the second subgroup property (i.e., normal subgroup)
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There exist situations where a permutable subgroup of a group is not normal.
- Omega-1 of odd-order class two p-group has prime exponent: For a -group of nilpotence class two with odd order, the set of elements of order forms a subgroup (along with the identity element).
Example of a group of prime power order
Let be an odd prime. Let be a group generated by , with the relations and . In other words, is a semidirect product of a cyclic group of order , by a cyclic group of order . Thus, has order .
We claim that the subgroup is a permutable subgroup of . For this, we need to show that permutes with every cyclic subgroup of .
Let's prove this. By the fact stated above, the set of elements of that have order (along with the identity element) form a subgroup of . This subgroup (we call it ) contains and . However, it cannot be bigger than the subgroup generated by and , which itself has order .
Now, if we pick any element inside , the cyclic subgroup generated by it clearly permutes with , because they're both inside an Abelian group of order . If we pick an element outside , then the cyclic subgroup it generates has order . Hence, that cyclic subgroup is maximal in and hence normal, so it commutes with . Thus, every cyclic subgroup commutes with , and we are done.