Alperin's conjecture on abelian-to-normal replacement for small index

History
This conjecture was made by Alperin in a paper in 1965.

Statement
Let $$k$$ be a nonnegative integer. Then, for all but finitely many primes $$p$$, any finite $$p$$-group that has an abelian subgroup of index $$p^k$$ also has an abelian normal subgroup of index $$p^k$$.