‣ GOuterGroup( E, N ) | ( function ) |
‣ GOuterGroup( ) | ( function ) |
Inputs a group \(E\) and normal subgroup \(N\). It returns \(N\) as a \(G\)-outer group where \(G=E/N\).
The function can be used without an argument. In this case an empty outer group \(C\) is returned. The components must be set using SetActingGroup(C,G), SetActedGroup(C,N) and SetOuterAction(C,alpha).
‣ GOuterGroupHomomorphismNC | ( global variable ) |
‣ GOuterGroupHomomorphismNC | ( global variable ) |
Inputs G-outer groups \(A\) and \(B\) with common acting group, and a group homomorphism phi:ActedGroup(A) --> ActedGroup(B). It returns the corresponding G-outer homomorphism PHI:A--> B. No check is made to verify that phi is actually a group homomorphism which preserves the G-action.
The function can be used without an argument. In this case an empty outer group homomorphism \(PHI\) is returned. The components must then be set.
Examples:
‣ GOuterHomomorphismTester( A, B, phi ) | ( function ) |
Inputs G-outer groups \(A\) and \(B\) with common acting group, and a group homomorphism phi:ActedGroup(A) --> ActedGroup(B). It tests whether phi is a group homomorphism which preserves the G-action.
The function can be used without an argument. In this case an empty outer group homomorphism \(PHI\) is returned. The components must then be set.
Examples:
‣ Centre( A ) | ( function ) |
Inputs G-outer group \(A\) and returns the group theoretic centre of ActedGroup(A) as a G-outer group.
‣ DirectProductGog( A, B ) | ( function ) |
‣ DirectProductGog( Lst ) | ( function ) |
Inputs G-outer groups \(A\) and \(B\) with common acting group, and returns their group-theoretic direct product as a G-outer group. The outer action on the direct product is the diagonal one.
The function also applies to a list Lst of G-outer groups with common acting group.
For a direct product D constructed using this function, the embeddings and projections can be obtained (as G-outer group homomorphisms) using the functions Embedding(D,i) and Projection(D,i).
Examples:
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