Prime power order implies not centerless
- Prime power order implies nilpotent: This follows from the result that a nontrivial group of prime power order is not centerless, the fact that a quotient of a group of prime power order also has prime power order, and by induction.
- Prime power order implies center is normality-large: This is a stronger version of the result stated on this page; the center is not just nontrivial, it intersects every nontrivial normal subgroup, nontrivially.
- Locally finite Artinian p-group implies not centerless: This is an attempt to weaken the hypothesis from finiteness, to weaker conditions.
Since both are groups of order a power of , the group being nontrivial is equivalent to the center being nontrivial -- either means that the two sides of the congruence are 0 mod .