Pi-Hall subgroups exist in pi-separable

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Suppose \pi is a set of primes and G is a finite group that is a \pi-separable group (see Pi-separable group (?)). Then the following are true:

  • G has a \pi-Hall subgroup.
  • G has a \sigma-Hall subgroup for \sigma = \pi \cup \{ q \} where q is a prime outside \pi.
  • G has a \sigma-Hall subgroup for \sigma = \{ p,q \} where p \in \pi and q \notin \pi.

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