# 3 Residual Functions

- `ResidualWrtFormation(` G`,` F`) O`

Let G be a finite solvable group and F a formation. Then `ResidualWrtFormation` returns the F-residual subgroup of G.

The following special cases have their own functions.

- `NilpotentResidual(` G`) A`

This is the last term of the descending central series of G.

- `PResidual(` G`,` p`) O`

This is the smallest normal subgroup of G whose index is a power of the prime p.

- `PiResidual(` G`,` primes`) O`

This is the smallest normal subgroup of G whose index is divisible only by primes in the list primes.

- `CoprimeResidual(` G`,` primes`) O`

This is the smallest normal subgroup of G whose index is divisible only by primes not in the list `primes`.

- `ElementaryAbelianProductResidual(` G`) A`

This is the smallest normal subgroup of G whose factor group is a direct product of groups of prime order.

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November 2000