Group Products

This chapter describes the various group product constructions that are possible in &GAP;.

At the moment for some of the products methods are available only if both factors are given in the same representation or only for certain types of groups such as permutation groups and pc groups when the product can be naturally represented as a group of the same kind.

&GAP; does not guarantee that a product of two groups will be in a particular representation. (Exceptions are and which are construction that makes sense only for permutation groups, see ).

&GAP; however will try to choose an efficient representation, so products of permutation groups or pc groups often will be represented as a group of the same kind again.

Therefore the only guaranteed way to relate a product to its factors is via the embedding and projection homomorphisms, see .

Direct Products

The direct product of groups is the cartesian product of the groups (considered as element sets) with component-wise multiplication.

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Semidirect Products

The semidirect product of a group N with a group G acting on N via a homomorphism \alpha from G into the automorphism group of N is the cartesian product G \times N with the multiplication (g, n) \cdot (h, m) = (gh, n^{{h^\alpha}}m).

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Subdirect Products

The subdirect product of the groups G and H with respect to the epimorphisms \varphi\colon G \rightarrow A and \psi\colon H \rightarrow A (for a common group A) is the subgroup of the direct product G \times H consisting of the elements (g,h) for which g^{\varphi} = h^{\psi}. It is the pull-back of the following diagram.

\matrix{ &&G&\cr &&\Big\downarrow&\varphi\cr H&\mathop{\longrightarrow}\limits^{\psi}&A&\cr }
G | phi psi V H ---> A
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Wreath Products

The wreath product of a group G with a permutation group P acting on n points is the semidirect product of the normal subgroup G^n with the group P which acts on G^n by permuting the components.

Note that &GAP; always considers the domain of a permutation group to be the points moved by elements of the group as returned by , i.e. it is not possible to have a domain to include fixed points, I.e. P = \langle (1,2,3) \rangle and P = \langle (1,3,5) \rangle result in isomorphic wreath products. (If fixed points are desired the wreath product G \wr T has to be formed with a transitive overgroup T of P and then the pre-image of P under the projection G \wr T \rightarrow T has to be taken.)

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Free Products

Let G and H be groups with presentations \langle X \mid R \rangle and \langle Y \mid S \rangle, respectively. Then the free product G*H is the group with presentation \langle X \cup Y \mid R \cup S \rangle. This construction can be generalized to an arbitrary number of groups.

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Embeddings and Projections for Group Products

The relation between a group product and its factors is provided via homomorphisms, the embeddings in the product and the projections from the product. Depending on the kind of product only some of these are defined.

returns the nr-th embedding in the group product P. The actual meaning of this embedding is described in the manual section for the appropriate product.
returns the (nr-th) projection of the group product P. The actual meaning of the projection returned is described in the manual section for the appropriate product.