Closure-proper prime divisor

This article defines a property of a prime divisor of an integer, one that is important/useful in the use of Sylow theory to study groups

Definition

Definition with symbols

A prime divisor $p$ of a number $N$ is said to be closure-proper if the $p$-Sylow-closure is a proper subgroup.

Relation with other properties

Stronger properties

• Sylow-unique prime divisor

Related properties

• Core-nontrivial prime divisor

Testing

To establish that a given prime divisor is closure-proper we could use the small-index subgroup technique. Suppose we are able to guarantee that for any group of order $N$, there is a subgroup of index $d$ where $d<p$. Then, we have a homomorphism from the group to $S_{d}$, with the property that all the elements in $p$-Sylow subgroups get mapped to the identity. Hence, the kernel of this homomorphism contains the $p$-Sylow closure.

However, this kernel is clearly contained inside the index $d$ subgroup (in fact, it is the normal core of the index $d$ subgroup). Hence, we conclude that the prime divisor $p$ is closure-proper.

The hard part is thus establishing the existence of subgroups whose index is less than one of the prime divisors. One way of doing this is as follows. Suppose we are forced that one of the Sylow numbers is less than one of the prime divisors. Then, since the Sylow number equals the index of the normalizer, we are done.

Presentations/talks on this

• Using Sylow theory in the classification of finite simple groups