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Gill, Olivia Jo (2011) Geometric and homological methods in group theory: constructing small group resolutions. Doctoral thesis, London Metropolitan University.
Given two groups K and H for which we have the free crossed resolutions, B* --> K and C* --> H respectively. Our aim is to construct a free crossed resolution, A* --> G, by way of induction on the degree n, for any semidirect product G = K x H.
First we show how to find a set Z1 of generators for the free group A1 and the corresponding unique epimorphism from the free group on those generators to the semidirect product. This gives us the I-dimensional free crossed resolution A1 --> G1 (see Proposition 4.1).
Next we define a set of generators Z2 that together with Z1, constitute a generating set for the free crossed module A2 --> A1. where δ2 is crossed module homomorphism. Proposition 4.1 together with this free crossed module δ2 : A2 --> A1 define a 2-dimensional free crossed resolution for A2 --> A1 --> G (see Proposition 4.9).
We then define an exact sequence A3 --> A2 --> A1, where A3, is an (A1/ δ2A2)-module on generating set Z3 with module homomorphism δ3 : A3 --> A2 defined on the generators. Proposition 4.11 says that we have a crossed complex of length 3, i.e., A3 --> A2 --> A1 --> G, where Imδ3 ⊆ Ker δ2.
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