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7. Computing with GAP

7.1: How do I perform binary operations on the elements of a group?

The group operation in GAP is always multiplication, written with *. This works e.g. for groups of permutations, of matrices, and for abstract groups given by generators and relations.

While you can construct a number of groups of permutations, they all have the same multiplication, in the sense that if two permutations p1 and p2 are both in two permutation groups, their product in each group will be the permutation p1*p2 given by applying p1 followed by p2.

In fact, this essentially works the other way round. GAP first defines elements, such as permutations, matrices or words in abstract generators, with their operations such as multiplication and inverse, then it allows you to define groups of such objects, using the objects multiplication as the group operation.

All the many algorithms, implemented in GAP for the investigation of the structure of a group, are written using this symbol * for the group multiplication.