Sylow implies order-dominating
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a finite group. That is, it states that in a Finite group (?), every subgroup satisfying the first subgroup property (i.e., Sylow subgroup (?)) must also satisfy the second subgroup property (i.e., Order-dominating subgroup (?)). In other words, every Sylow subgroup of finite group is a order-dominating subgroup of finite group.
View all subgroup property implications in finite groups View all subgroup property non-implications in finite groups View all subgroup property implications View all subgroup property non-implications
The subgroup property of being a Sylow subgroup is stronger than, or implies, the subgroup property of being an order-dominating subgroup: for any subgroup whose order divides the order of the Sylow subgroup, some conjugate of that subgroup is contained in the given Sylow subgroup.
Statement with symbols
This statement is a part of Sylow's theorem.
- Sylow implies order-conjugate: This follows from the fact that order-dominating implies order-conjugate in co-Hopfian: in particular, in a finite group, any order-dominating subgroup is order-conjugate. In other words, any two -Sylow subgroups are conjugate.
Proof using coset spaces
Given: a finite group, a -Sylow subgroup, and a -group. Suppose where is the order of and is relatively prime to (So, .
To prove: There exists such that .
Proof: We prove this through a series of observations:
- naturally acts (on the left) on the left coset space of .
- Since is a subgroup of , also acts on the left coset space of
- The left coset space of has cardinality , which is relatively prime to . Hence, under the action of (which is a -group) on this set, there is at least one fixed point. Let the fixed point be .
- We have . This gives us: , and hence . Thus, a conjugate of is contained inside .