The factor group \ Q / A by a proper subgroup A is a divisible group, hence certainly not finitely generated, hence has a proper non-trivial subgroup, which gives rise to a subgroup and ideal containing A.
22.
Abelian surfaces are sometimes excluded from the classification of being Calabi & ndash; Yau, as their holonomy ( again the trivial group ) is a proper subgroup of SU ( 2 ), instead of being isomorphic to SU ( 2 ).
23.
For instance, for every cardinal " d " there exist torsion-free abelian groups of rank " d " that are indecomposable, i . e . cannot be expressed as a direct sum of a pair of their proper subgroups.
24.
Hypercentral groups enjoy many properties of nilpotent groups, such as the "'normalizer condition "'( the normalizer of a proper subgroup properly contains the subgroup ), elements of coprime order commute, and Sylow " p "-subgroups.
25.
It is not true in general that every automorphism of has the above form . ( That is, is often a proper subgroup of . ) For example, if is any group, then there exists an automorphism of that switches the two factors, i . e.
26.
A Fischer subgroup is a maximal nilpotent subgroup, but not every maximal nilpotent subgroup is a Fischer subgroup : again the nonabelian group of order six provides an example as every non-identity proper subgroup is a maximal nilpotent subgroup, but only the subgroup of order three is a Fischer subgroup.
27.
In 1982 Ol'shanskii was able to strengthen his results to establish existence, for any sufficiently large prime number " p " ( one can take " p " > 10 75 ) of a finitely generated infinite group in which every nontrivial proper subgroup is a cyclic group of order " p ".
28.
Any of the three proper subgroups of V ( all of which have order 2 ) is normal in V; fixing one of these subgroups, any of the other two remaining ( proper ) subgroups complements it in V, but none of these three subgroups of V is a conjugate of any other one, because V is Abelian.
29.
The Hawaiian earring " H " has the free group of countably infinitely many generators as a proper subgroup of its fundamental group . " G " contains additional elements, which arise from loops whose image is not contained in finitely many of the Hawaiian earring's circles; in fact, some of them are surjective.
30.
Specifically, for " n " = 2 ( a projective line ), all points are collinear, so the collineation group is exactly the symmetric group of the points of the projective line, and except for "'F "'2 and "'F "'3 ( where PGL is the full symmetric group ), PGL is a proper subgroup of the full symmetric group on these points.
