GAP (HAP commands) - Chapter 24: Cat-1-groups

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24 Cat-1-groups
24.1
24.1-1 AutomorphismGroupAsCatOneGroup
24.1-2 HomotopyGroup
24.1-3 HomotopyModule
24.1-4 QuasiIsomorph
24.1-5 ModuleAsCatOneGroup
24.1-6 MooreComplex
24.1-7 NormalSubgroupAsCatOneGroup
24.1-8 XmodToHAP

### 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 \(G\rightarrow Aut(G)\).

Examples: 1 , 2 , 3 , 4 , 5

##### 24.1-2 HomotopyGroup

`‣ HomotopyGroup`( C, n ) | ( function )

Inputs a cat-1-group \(C\) and an integer n. It returns the \(n\)th homotopy group of \(C\).

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7

##### 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.

Examples: 1

##### 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 \(C\rightarrow D\) that induces isomorphisms on homotopy groups.

This function was implemented by Le Van Luyen.

Examples: 1 , 2 , 3

##### 24.1-5 ModuleAsCatOneGroup

`‣ ModuleAsCatOneGroup` | ( global variable )

Inputs a group \(G\), an abelian group \(M\) and a homomorphism \(\alpha\colon G\rightarrow Aut(M)\). It returns the Cat-1-group \(C\) corresponding th the zero crossed module \(0\colon M\rightarrow G\).

Examples:

##### 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).

Examples:

##### 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 \( N\rightarrow G\).

Examples:

##### 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.

Examples: 1

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