Core for a set of prime divisors

Statement

Suppose $G$ is a finite group and $π$ is a set of prime numbers (we may without loss of generality assume that $π$ only includes primes that divide the order of $G$ ). The $π$ -core of $G$ , denoted $O_{π} (G)$ is defined in the following equivalent ways:

It is the subgroup generated by all normal $π$ -subgroups, i.e., all normal subgroups for which the prime divisors of their order are in $π$ .
It is the subgroup generated by all characteristic $π$ -subgroups, i.e., all characteristic subgroups for which the prime divisors of their order are in $π$ .
It is the unique largest normal $π$ -subgroup of $G$ .
It is the unique largest characteristic $π$ -subgroup of $G$ .
It is the subgroup generated by $O_{p} (G)$ for all $p \in π$ , where $O_{p} (G)$ is the normal core of any $p$ -Sylow subgroup, or equivalently, is the largest normal $p$ -subgroup.