GAP (HAP) - Chapter 20: Cat-1-groups

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20 Cat-1-groups

### 20 Cat-1-groups

`AutomorphismGroupAsCatOneGroup(G)` Inputs a group G and returns the Cat-1-group C corresponding to the crossed module G→ Aut(G).
`HomotopyGroup(C,n)` Inputs a cat-1-group C and an integer n. It returns the nth homotopy group of C.
`HomotopyModule(C,2)` Inputs a cat-1-group C and an integer n=2. It returns the second homotopy group of C as a G-module (i.e. abelian G-outer group) where G is the fundamental group of C.
`QuasiIsomorph(C)` Inputs a cat-1-group C and returns a cat-1-group D for which there exists some homomorphism C→ D that induces isomorphisms on homotopy groups. This function was implemented by Le Van Luyen.
`ModuleAsCatOneGroup(G,alpha,M)` Inputs a group G, an abelian group M and a homomorphism α: G→ Aut(M). It returns the Cat-1-group C corresponding th the zero crossed module 0: M→ G.
`MooreComplex(C)` Inputs a cat-1-group C and returns its Moore complex as a G-complex (i.e. as a complex of groups considered as 1-outer groups).
`NormalSubgroupAsCatOneGroup(G,N)` Inputs a group G with normal subgroup N. It returns the Cat-1-group C corresponding th the inclusion crossed module N→ G.
`XmodToHAP(C)` Inputs a cat-1-group C obtained from the Xmod package and returns a cat-1-group D for which IsHapCatOneGroup(D) returns true. It returns "fail" id C has not been produced by the Xmod package.

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