The GAP 4 package FORMAT provides functions to compute with formations of finite solvable groups. In addition to tools for constructing and combining formations, the package contains functions to compute F-residual subgroups and to construct F-normalizers and F-covering subgroups determined by locally defined formations. System normalizers and Carter subgroups are available as special cases, and the F-normalizer functions also apply to the computation of complements. The corresponding algorithms, together with applications and a complexity analysis, are described in EW.
The package permits the computation of formation-theoretic subgroups not only for a number of classical formations, such as nilpotent, supersolvable or p-length 1 groups, but for other formations that the user may define. It also allows computation with classes of finite solvable groups defined by normal subgroup functions (see DH, pages 395 ff). Attention may be restricted to the subgroups of a single group, a feature that has applications in the computation of complements to elementary abelian normal subgroups in finite solvable groups (see EW). An example of such an application is given in Section Other Applications.
This documentation contains only a brief account of the main formation-theoretic ideas. For a much more complete treatment we refer the reader to DH. Fundamental ideas of formation theory are described in G and CH.
In the following sections we first describe the GAP definition of a formation and the examples of standard formations that are included in the package. We also present some functions that obtain new formations from ones already defined or that modify defined formations slightly. (See Section Formations in GAP.)
Then we describe functions that compute formation-theoretic subgroups of finite solvable groups (see Sections Residual Functions, FNormalizers and Covering Subgroups).
Finally we provide examples from a GAP session (see Sections Formation Examples and Other Applications) to illustrate the functions in the package.
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