Magma provides facilities for constructing and working with automorphism groups of various objects. In this chapter we describe the machinery provided in Magma for groups of automorphisms in the case of groups.
An automorphism of a group G is a bijective homomorphism from G to itself. The set of all automorphisms of G forms a group U known as the automorphism group of G. A subgroup A of U will be referred to as a group of automorphisms of G. The group G is called the base group of a group of automorphisms A and we say that A acts on G. Each Magma automorphism group A stores, as part of its data structure, a generating set for its base group, and each automorphism of A is described by its action on these generators.
The full group of automorphisms may be found using an algorithm that proceeds as follows: A series of characteristic subgroups 1 = Nr < Nr - 1 < ... < N1=L < G is constructed for the given group G, such that each Ni/Ni + 1 is elementary abelian and such that G/L has no non-trivial soluble normal subgroup. The automorphism group is found for each of the associated factor groups of G, starting with the top factor G/L and lifting through each layer Ni/Ni + 1 in turn, until we finally have the automorphism group for G itself. The general algorithm for a non-soluble group is described in Cannon and Holt [CH03]. More specialised versions are described by Eick, Leedham-Green and O'Brien [ELGO02] (p-groups) and Smith [Smi94] (soluble groups). This general class of algorithms will be referred to collectively as lifting algorithms.
When G is a non-soluble permutation or matrix group, the algorithm relies on a database of automorphism groups for the non-cyclic simple factors of G, hence the non-abelian composition factors of G must belong to a restricted list. In V2.20 this list includes all simple groups of order at most 1.6times107, the alternating groups of degree at most 2499, L2(q), L3(q), L4(q), L5(q), L6(q) and L7(q) for all q, U3(q) for all q, U4(q) for all q, S4(q) for all q, Ld(2) for d ≤14, and the following groups: U5(3), U6(2), U7(2), U8(2), S6(3), S6(4), S6(5), S8(2), S8(3), S10(2), S12(2), O∓8(2), O∓8(3), O∓8(4), O∓10(2), O∓12(2), O7(3), O7(5), O9(3), G2(4), G2(5), ()3D4(2), ()2F4(2)', Co2, Co3, He, Fi22, Ru, Suz, ON. The list is being extended regularly.
An automorphism group A of G is represented as a set of homomorphisms of G into itself. We shall refer to this as the mappings representation of A. The full automorphism group is also returned as a finitely presented group and, in addition, it is also possible to construct a permutation representation of the automorphism.
The family of all groups of automorphisms forms a category. The objects are the automorphism groups and the morphisms are group homomorphisms. The Magma designation for this category of automorphism groups is GrpAuto.