Let be a normal -subgroup (not necessarily Sylow) of a finite group.
- Prove that is contained in every Sylow -subgroup of .
- If is another normal -subgroup of , prove that is also a normal -subgroup of .
- Let , and define . Prove that is the -greatest normal -subgroup of . Prove further that .
- Prove that . (That is, has no nontrivial normal -subgroups.)
- Let be a normal -subgroup. Now for some Sylow -subgroup by Sylow’s Theorem. Also by Sylow, if is a second Sylow -subgroup, then for some . Then , so that .
- Let and be normal -subgroups. If is a Sylow -subgroup, then by part (1). Then is a -subgroup, and moreover is normal.
- We need to show three things: normalcy, -groupitude, and -greatestness.
- Let . Then . Thus is normal.
- If is some Sylow -subgroup of , then for each . Thus , and we have ; thus is a -subgroup.
- Suppose is some other normal -subgroup; then by definition. Thus is the -greatest normal -subgroup of .
We showed that for all Sylow -subgroups , hence . To see the other inclusion, note that for each , conjugation by permutes the Sylow -subgroups of . That is, = ; hence is a normal subgroup of , hence is contained in .
- Suppose is a nontrivial normal -subgroup. By the Lattice Isomorphism Theorem, for some normal subgroup with . By Lagrange’s Theorem, we have , so that . Then is a normal -subgroup of , and we have . But then is trivial, a contradiction. So no such subgroup exists.