1%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2%% 3%W intro.tex FORMAT documentation B. Eick and C.R.B. Wright 4%% 5%% 10-31-11 6 7%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8%\def\FORMAT{{\sf FORMAT}} 9\def\F{{\cal{F}}} 10\Chapter{Introduction to FORMAT} 11 12\index{Format} 13 14The {\GAP} package {\FORMAT} provides functions to compute with formations 15of finite solvable groups. In addition to tools for constructing and 16combining formations, the package contains functions to compute 17$\F$-residual subgroups and to construct $\F$-normalizers and 18$\F$-covering subgroups determined by locally defined formations. 19System normalizers and Carter subgroups are available as special cases, 20and the $\F$-normalizer functions also apply to the computation of 21complements. The corresponding algorithms, together with applications 22and a complexity analysis, are described in~\cite{EW}. 23 24The package permits the computation of formation-theoretic subgroups 25not only for a number of classical formations, such as nilpotent, 26supersolvable or $p$-length 1 groups, but for other formations that the 27user may define. It also allows computation with classes of 28finite solvable groups defined by normal subgroup functions (see 29\cite{DH}, pages 395~ff). Attention may be restricted to the 30subgroups of a single group, a feature that has applications 31in the computation of complements to elementary abelian normal subgroups 32in finite solvable groups (see \cite{EW}). An example of such an 33application is given in Section~"Other Applications". 34 35This documentation contains only a brief account of the main 36formation-theoretic ideas. For a much more complete treatment we 37refer the reader to \cite{DH}. Fundamental ideas of formation theory are 38described in \cite{G} and \cite{CH}. 39 40In the following sections we first describe the {\GAP} definition of a 41formation and the examples of standard formations that are included in 42the package. We also present some functions that obtain new formations 43from ones already defined or that modify defined formations slightly. 44(See Section~"Formations in GAP".) 45 46Then we describe functions that compute formation-theoretic subgroups 47of finite solvable groups (see Sections "Residual Functions", 48"FNormalizers" and~"Covering Subgroups"). 49 50Finally we provide examples from a {\GAP} session (see Sections~"Formation 51Examples" and "Other Applications") to illustrate the functions in the package. 52 53