%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%W  resid.tex       FORMAT documentation        B. Eick and C.R.B. Wright
%%
%%  12-28-99
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Chapter{Residual Functions}

%\index{Residual Functions}

\bigskip

\> 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.

\bigskip

\> 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.


