Core for a set of prime divisors

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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.