# Sylow not implies local divisibility-closed

From Groupprops

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) neednotsatisfy 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

## Statement

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 .

## Proof

`Further information: symmetric group:S4, D8 in S4`

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 .