Core for a prime divisor

Definition

Definition with symbols

Let $G$ be a finite group of order $N$ and let $p$ be a prime divisor of $N$. Then, the $p$-core of $G$, donated by $O_{p}(G)$ is defined as the following subgroup:

• It is the intersection of all $p$-Sylow subgroups
• It is the subgroup generated by all normal $p$-Sylow subgroups
• It is the subgroup generated by all characteristic $p$-Sylow subgroups

Relation with other subgroup-defining functions

Fitting subgroup

The Fitting subgroup is defined as the subgroup generated by all normal nilpotent subgroups. In the case of a finite group, the Fitting subgroup is simply the product of all $p$-cores for prime divisors $p$.

Related properties of prime divisors

Core-trivial and core-nontrivial

For a given group $G$, a prime divisor $p$ is said to be core-trivial if the $p$-core is trivial, and is said to be core-nontrivial if the $p$-core is nontrivial.

If we are not given the actual group $G$ but only the order of $G$, it may happen that for some groups with that order, the core is trivial while for others it is nontrivial. For instance, in the cyclic group of that order, the core is the whole Sylow subgroup, whereas if there is a simple group of that order, the core is trivial.

A prime divisor of a number is termed core-nontrivial if for any group of that order, the core for that prime divisor is nontrivial.