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.