Let , the free group generated by and a subgroup.

Both and the group generated by are primitive.

The unitary group generated by i 'H' corresponds to the time evolution of 'S'.

It is the group generated by all the normal subgroups of G that lie in H.

The free abelian group generated by this basis is commonly denoted as .

There are also non-abelian groups generated by rotations around different axes.

The finite groups generated in this way are examples of Coxeter groups.

Informally is the group generated by the set such that for all .

The groups generated by a reflection leave a point fixed.

All other subgroups of 2T are cyclic groups generated by the various elements, with orders 3, 4, and 6.