*All functions defined in this package deal only with finite groups.* Moreover, some of the functions assume that the orders of all subgroups are easily computable and that the decomposition of the order of a group as a product of prime numbers can be done in a reasonable time.

The package **PERMUT** contains some functions to deal with permutability in finite groups. It includes functions to test some subgroup embedding properties related to permutability, like permutability or Sylow permutability. It also includes some functions to check whether a group belongs to the classes of T-groups, PT-groups, and PST-groups, which are the classes of groups in which normality, permutability, and Sylow permutability, respectively, are transitive. These properties and classes of groups have been widely studied during the last years. Most of them are described in [BBERA10].

The algorithms for T-groups, PT-groups, and PST-groups of this package use some interesting local descriptions of groups in these classes, that is, given in terms of some information related to the primes \(p\) dividing their order, usually by looking at \(p\)-subgroups or \(p\)-chief factors. These characterisations show that the only difference between all three classes of groups in the soluble universe corresponds to the Sylow structure. Nevertheless, for the sake of completeness, we also provide functions that use directly the definition of these classes. In the case of T-groups and PST-groups, as well as for soluble PT-groups, we reduce the test to subnormal subgroups of defect \(2\) (see [BBERR07], [BBERR09], and [BBBC+09]). Of course, to do this we must introduce some functions to check whether two subgroups permute and whether a subgroup is permutable or S-permutable.

Some of the definitions of group-related concepts appear more than once in this manual, in the description of different functions. Although these repetitions may seem unnecessary when reading the whole manual, we hope that they benefit users who read the online help in **GAP**.

In order to obtain easily counterexamples which show that a group or a subgroup does not satisfy a certain property, we have introduced what we have called "One" functions, which store such counterexamples. In some cases, the property can be checked by proving that these counterexamples do not exist.

This package requires the **Format** package by B. Eick and C. R. B. Wright (see [EW03]), because it uses the functions `PResidual`

(FORMAT: PResidual) and `SystemNormalizer`

(FORMAT: SystemNormalizer), which are defined there. Some of the examples in this manual use the library of groups of small order.

The mathematical foundations of the algorithms presented in this package have been described in [BBCLER13].

The authors acknowledge the support of the grants MTM2010-19938-C03-01 and MTM2014-54707-C3-1-P funded by the *Ministerio de Economía y Competitividad*, Spanish Government (all authors), PGC2018-095140-B-I00, funded by MCIN/AEI/10.13039/501100011033 and by ``ERDF A way of making Europe'' (all authors), the grant PROMETEO/2017/057 funded by GVA/10.13039/501100003359 (A. Ballester-Bolinches and R. Esteban-Romero), the grant 11271085 from National Natural Science Foundation of China (A. Ballester-Bolinches) and the predoctoral grant AP2010-2764 from the *Ministerio de Educación*, Spanish Government (E. Cosme-Llópez). The authors are also indebted to the members of the **GAP** council, especially Leonard Soicher, Alice Niemeyer, Max Horn, and Alexander Konovalov, as well as to the anonymous referees, for their comments which have helped us to improve the package and its documentation.

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