P-group is either absolutely regular or maximal class or has normal subgroup of exponent p and order p^p

Statement
Suppose $$p$$ is a prime number and $$P$$ is a finite $$p$$-group, i.e., a group of prime power order where the underlying prime is $$p$$. Then, at least one of the following holds:


 * 1) $$P$$ is an fact about::absolutely regular p-group.
 * 2) $$P$$ is a fact about::maximal class group.
 * 3) $$P$$ contains a normal subgroup $$N$$ of exponent p and order $$p^p$$.

Related facts

 * Group of exponent p and order greater than p^p is not embeddable in a maximal class group
 * Mann's replacement theorem for subgroups of prime exponent

Journal references

 * , Theorem 1.1
 * , Theorem 7.6