Sylow not implies local divisibility-closed
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., Sylow subgroup) need not satisfy the second subgroup property (i.e., local divisibility-closed subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
Get more facts about Sylow subgroup|Get more facts about local divisibility-closed subgroup
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property Sylow subgroup but not local divisibility-closed subgroup|View examples of subgroups satisfying property Sylow subgroup and local divisibility-closed subgroup
It is possible to have a finite group , a prime number , and a -Sylow subgroup of such that is not a local divisibility-closed subgroup of . In particular, we can set our example so that there are elements of that have roots in but such that none of the roots is in .
Consider the following:
- is symmetric group:S4, acting on the set .
- is the subgroup D8 in S4, i.e., the 2-Sylow subgroup. Explicitly, it is the subset:
The element has square roots and in , but neither of these square roots is in .