Template:Random interesting fact

The union of all conjugates of a proper subgroup in a finite group can never be the whole group (FACT): And this breaks down for infinite groups.@@@The product of two proper conjugate subgroups of a group (not necessarily finite) is proper (FACT): However, the subgroup they generate could be the whole group.@@@Nilpotent implies center is normality-large (FACT): In a nilpotent group, the center intersects every nontrivial normal subgroup nontrivially.@@@Characteristic of normal implies normal (FACT): And characteristicity is also the tightest such property.@@@Supersolvable implies every nontrivial normal subgroup contains a cyclic normal subgroup (FACT): This, incidentally, helps prove that maximal among abelian normal implies self-centralizing in supersolvable.