In this paper we describe a share package XMOD [1] of functions for computing with finite, permutation crossed modules, their morphisms and derivations; cat1- groups, their morphisms and their sections, written using the GAP group theory programming language. | Turk J Math 24 (2000) , 121 – 128. ¨ ITAK ˙ c TUB Some Results on Derivation Groups Murat Alp Abstract In this paper we describe a share package XMOD [1] of functions for computing with finite, permutation crossed modules, their morphisms and derivations; cat1 groups, their morphisms and their sections, written using the GAP [5] group theory programming language. We also give some mathematical results for derivations. These results are suggested by the output produced by the XMOD [1] package. Key Words: Crossed modules, derivation, whitehead multiplication. 1. Introduction A starting point for this paper was to consider the possibility of implementing functions for doing calculations with crossed modules, derivations, actor crossed modules, cat1groups, sections, induced crossed modules and induced cat1-groups in GAP [5]. We should first explain the importance of crossed modules. The general points are: • crossed modules may be thought of as 2-dimensional groups; • a number of phenomena in group theory are better seen from a crossed module point of view; • crossed modules occur geometrically as π2 (X, A) → π1 A when A is a subspace of X or as π1 F → π1 E where F → E → B is a fibration; • crossed modules are usefully related to forms of double groupoids. 1991 A. M. S. C.: & 13D99, 16A99, 17B99, 17D99, 18D35. 121 ALP Particular constructions, such as induced crossed modules, are important for the applications of the 2-dimensional Van-Kampen Theorem of Brown and Higgins [2], and so for the computation of homotopy 2-types. For all these reasons, the facilitation of the computations with crossed modules should be advantageous. It should help to solve specific problems, and it should make it easier to construct examples and see relations with better known theories. The powerful computer algebra system GAP provides a high level programming language with several advantages for the coding of new mathematical structures. The GAP system has been developed over the .
