### 24 Cat-1-groups

#### 24.1

##### 24.1-1 AutomorphismGroupAsCatOneGroup
 ‣ AutomorphismGroupAsCatOneGroup( G ) ( function )

Inputs a group G and returns the Cat-1-group C corresponding to the crossed module G→ Aut(G).

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##### 24.1-2 HomotopyGroup
 ‣ HomotopyGroup( C, n ) ( function )

Inputs a cat-1-group C and an integer n. It returns the nth homotopy group of C.

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##### 24.1-3 HomotopyModule
 ‣ HomotopyModule( C, 2 ) ( function )

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.

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##### 24.1-4 QuasiIsomorph
 ‣ QuasiIsomorph( C ) ( function )

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 .

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##### 24.1-5 ModuleAsCatOneGroup
 ‣ ModuleAsCatOneGroup ( global variable )

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.

##### 24.1-6 MooreComplex
 ‣ MooreComplex( C ) ( function )

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).

##### 24.1-7 NormalSubgroupAsCatOneGroup
 ‣ NormalSubgroupAsCatOneGroup( G, N ) ( function )

Inputs a group G with normal subgroup N. It returns the Cat-1-group C corresponding th the inclusion crossed module N→ G.

##### 24.1-8 XmodToHAP
 ‣ XmodToHAP( C ) ( function )

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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