36.2 Creating Groups

Group( gen, ... ) is the group generated by the arguments gen, ...

If the only argument gens is a list that is not a matrix then Group( gens ) is the group generated by the elements of that list.

If there are two arguments, a list gens and an element id, then Group( gens, id ) is the group generated by the elements of gens, with identity id.

Note that the value of the attribute GeneratorsOfGroup need not be equal to the list gens of generators entered as argument. Use GroupWithGenerators if you want to be sure that the argument gens is stored as value of GeneratorsOfGroup. If this is required, see GroupWithGenerators(GroupWithGenerators).

gap> g:=Group((1,2,3,4),(1,2));
Group( [ (1,2,3,4), (1,2) ] )

GroupWithGenerators returns the group G generated by the list gens. If a second argument id is present then this is stored as the identity element of the group. The value of the attribute GeneratorsOfGroup of G is equal to gens.

returns a list of generators of the group G. If G has been created by the command GroupWithGenerators (see GroupWithGenerators), with argument gens, then the list returned by GeneratorsOfGroup will be equal to gens.

gap> g:=GroupWithGenerators([(1,2,3,4),(1,2)]);
Group( [ (1,2,3,4), (1,2) ] )
gap> GeneratorsOfGroup(g);
[ (1,2,3,4), (1,2) ]

if the elements of the collection D form a group the command returns this group, otherwise it returns fail.

gap> AsGroup([(1,2)]);           
fail
gap> AsGroup([(),(1,2)]);
Group( [ (1,2) ] )

returns the conjugate group of G, obtained by applying the conjugating element obj. To form a conjugate (group) by any object acting via ^, one can use the infix operator ^.

gap> ConjugateGroup(g,(1,5));
Group( [ (2,3,4,5), (2,5) ] )

A group is a magma-with-inverses (see IsMagmaWithInverses) and associative (see IsAssociative) multiplication.

IsGroup tests whether the object obj fulfills these conditions, it does not test whether obj is a set of elements that forms a group under multiplication; use AsGroup (see AsGroup) if you want to perform such a test. (See Categories for details about categories.)

gap> IsGroup(g);
true

is the info class for the generic group theoretic functions (see Info Functions).

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