Core for a set of prime divisors

Statement

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

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