Documentation

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 g₁, ..., g_n, and that φ(g_i)=h_i for each i, where φ 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 -> h_i (order not important);
(c)
h₁, ..., 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 {g₁, ..., 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 g₁, ..., g_n, and that φ(g_i)=h_i for each i, where φ 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 -> h_i (order not important);
(c)
h₁, ..., 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 {h₁, ..., 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 `GrpAb_Homomorphisms (H75E15)`

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

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

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

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]);