Homomorphisms#

Two functions are provided to construct homomorphisms or isomorphisms from one group into another, where either of the groups, or both, may be generic abelian groups.

hom< A -> B | L>: Grp, Grp, List -> Map#

Given groups \(A\) and \(B\), construct a homomorphism from \(A\) to \(B\) as defined by the extension \(L\). If one or both of \(A\) and \(B\) are generic abelian groups this works as usual, with one minor difference as explained below. Suppose that the generators of \(A\) are \(g_1, \dots, g_n\), and that \(\phi(g_i)=h_i\) for each \(i\), where \(\phi\) is the homomorphism one wishes to construct. The list \(L\) as required by the constructors must be one of the following:

(a): a list of the \(n\) 2-tuples \(< g_i, h_i >\) (order not important);
(b): a list of the \(n\) arrow-pairs \(g_i \rightarrow h_i\) (order not important);
(c): \(h_1, \dots, h_n\) (order is important).

If \(A\) is a generic abelian group this rule is relaxed somewhat in the following sense: If \(L\) is a list of \(n\) 2-tuples or of \(n\) arrow-pairs, the elements \(g_i\) need not be the defining generators of \(A\). The only requirement is that the set \(\{g_1, \ldots, g_n\}\) does actually generate the whole of \(A\).

Homomorphism(A, B, X, Y): Grp, Grp, [ GrpElt ], [ GrpElt ] -> Map#

Creates a homomorphism from \(A\) into \(B\) as given by the mapping of \(X\) into \(Y\). The arguments \(A\) and \(B\) may be any type of group, including of course, generic abelian groups.

The function Homomorphism does not require the elements of the argument \(X\) to be generators of \(A\) as given by Generators(A), so it allows more freedom when creating a homomorphism. If, however, these elements fail to generate the whole of \(A\) then the subsequent map application will fail.

iso< A -> B | L>: Grp, Grp, List -> Map#

Given groups \(A\) and \(B\), construct an isomorphism from \(A\) to \(B\) as defined by the extension \(L\). If one or both of \(A\) and \(B\) are generic abelian groups this works as usual, with one minor difference as explained below. Suppose that the generators of \(A\) are \(g_1, \dots, g_n\), and that \(\phi(g_i)=h_i\) for each \(i\), where \(\phi\) is the isomorphism one wishes to construct. The list \(L\) as required by the constructors must be one of the following:

(a): a list of the \(n\) 2-tuples \(< g_i, h_i >\) (order not important);
(b): a list of the \(n\) arrow-pairs \(g_i \rightarrow h_i\) (order not important);
(c): \(h_1, \dots, h_n\) (order is important).

If \(A\) is a generic abelian group this rule is relaxed somewhat in the following sense: If \(L\) is a list of \(n\) 2-tuples or of \(n\) arrow-pairs, the elements \(g_i\) may not necessarily be generators of \(A\) as given by the function Generators(A). The only requirement is that the set \(\{h_1, \ldots, h_n\}\) does actually generate the whole of \(B\).

Isomorphism(A, B, X, Y): Grp, Grp, [ GrpElt ], [ GrpElt ] -> Map#

Creates a isomorphism from \(A\) into \(B\) as given by the mapping of \(X\) into \(Y\). The arguments \(A\) and \(B\) can be any type of group, including of course, generic abelian groups.

The function Isomorphism does not require the elements of the argument \(X\) to be generators of \(A\) as given by Generators(A), so it allows more freedom when creating an isomorphism. If, however, these elements fail to generate the whole of \(A\) then the subsequent map application will fail.

Example: Homomorphisms (ex-32bea9)#

Recall that we defined the subgroups \(GH1_{Zm}\) and \(GH2_{Zm}\) of \(G\) as:

> GH1_Zm;
Generic Abelian Group over
Residue class ring of integers modulo 34384
Abelian Group isomorphic to Z/6 + Z/612
Defined on 2 generators in supergroup G:
  GH1_Zm.1 = G.2 + G.3
  GH1_Zm.2 = G.4
Relations:
  6*GH1_Zm.1 = 0
  612*GH1_Zm.2 = 0
> GH2_Zm;
Generic Abelian Group over
Residue class ring of integers modulo 34384
Abelian Group isomorphic to Z/2 + Z/2 + Z/6 + Z/612
Defined on 4 generators in supergroup G:
  GH2_Zm.1 = G.1
  GH2_Zm.2 = G.2
  GH2_Zm.3 = G.3
  GH2_Zm.4 = G
Relations:
  2*GH2_Zm.1 = 0
  2*GH2_Zm.2 = 0
  6*GH2_Zm.3 = 0
  612*GH2_Zm.4 = 0

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We construct the homomorphism

> h := hom<GH1_Zm -> GH2_Zm | GH2_Zm.1, GH2_Zm.2 >;
> h(GH1_Zm);
Generic Abelian Group over
Residue class ring of integers modulo 34384
Abelian Group isomorphic to Z/2 + Z/2
Defined on 2 generators in supergroup GH2_Zm:
  $.1 = GH2_Zm.2
  $.2 = GH2_Zm.1
Relations:
  2*$.1 = 0
  2*$.2 = 0

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but we cannot construct the isomorphism

> i := iso<GH1_Zm -> GH2_Zm | GH2_Zm.1, GH2_Zm.2 >;

>> i := iso<GH1_Zm -> GH2_Zm | GH2_Zm.1, GH2_Zm.2 >;
           ^
Runtime error in map< ... >: Images do not generate the (whole) codomain

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An alternative way of creating the homomorphism \(h\) would be

> h := Homomorphism(GH1_Zm, GH2_Zm, Generators(GH1_Zm), [GH2_Zm.1, GH2_Zm.2]);

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