‣ 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 , 6
‣ 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 , 8
‣ 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.
‣ 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.
‣ 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:
‣ 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:
‣ 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:
‣ 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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