GAP:GeneratorsOfGroup

Function type
The function GeneratorsOfGroup takes as input a group and outputs a list of elements.

Behavior
GAP stores groups in different forms (through finite presentations, polycyclic series, or permutation representations). In each of these forms, there is included a set of generators. The command GeneratorsOfGroup returns this set of generators as a list.


 * For a group stored using a polycyclic series, the generators are by default labeled $$f1, f2, \dots$$. If further groups are constructed starting from such a group, the generators are described in terms of these original generators. Note that for any finite solvable group, the SmallGroup command stores the group as a polycyclic series.
 * For a group stored in the form of a presentation, the generators are again labeled $$f1, f2, \dots$$.
 * For a group stored in the form of a faithful permutation representation, the generators are stored as permutations: the elements of the symmetric group that generate the given group.

Note that the stored generating sets may be different for isomorphic groups because of differences in the way they are constructed.

Examples of usage
gap> GeneratorsOfGroup(SymmetricGroup(2)); [ (1,2) ] gap> GeneratorsOfGroup(SymmetricGroup(3)); [ (1,2,3), (1,2) ] gap> GeneratorsOfGroup(SmallGroup(6,1)); [ f1, f2 ] gap> IsomorphismGroups(SymmetricGroup(3),SmallGroup(6,1)); [ (1,2,3), (1,2) ] -> [ f2, f1*f2 ] gap> GeneratorsOfGroup(CyclicGroup(6)); [ f1, f2 ] gap> GeneratorsOfGroup(SmallGroup(60,5)); [ (1,2,3,4,5), (1,2,3) ] gap> GeneratorsOfGroup(AlternatingGroup(5)); [ (1,2,3,4,5), (3,4,5) ] gap> GeneratorsOfGroup(FreeGroup(5)); [ f1, f2, f3, f4, f5 ]

In the first two commands, the groups are described as permutation groups, so the generating sets are given as sets of permutations. In the third example, the group is created using the SmallGroup command, and is described using a polycyclic series. The next command tests for an isomorphism between the symmetric group on three elements and the small group with ID $$(6,1)$$ -- the answer turns out to be affirmative, and an explicit isomorphism is provided in terms of what happens to the generators. The next command finds a generating set for the cyclic group of order six. Note that this generating set is the correct generating set for a polycyclic series -- it is not the smallest possible generating set.

The next two commands compute generating sets for the alternating group using two descriptions: one, using the group ID, and the other, using the AlternatingGroup command. Both descriptions give generating sets as permutations; however, the specific generating sets differ in appearance.

The final command computes the generating set for the free group on a set of size five.