%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %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.