Alperin's theorem on non-existence of abelian subgroups of large prime power order for odd prime

History
This result was shown in a paper by Alperin in 1965.

Statement
Let $$p$$ be an odd prime and $$k \ge 4$$. Then, there exists a group of order $$p^{3k - 7}$$ that does not contain any abelian subgroup of order $$p^k$$.

Related facts

 * Alperin's theorem on non-existence of abelian subgroups of large prime power order for prime equal to two