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

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

Statement
There exists a group of order $$2^{50}$$ that does not contain any abelian subgroup of order $$2^{25}$$.

Related facts

 * Alperin's theorem on non-existence of abelian subgroups of large prime power order for odd prime
 * Existence of abelian normal subgroups of small prime power order
 * Abelian-to-normal replacement theorem for prime-cube order
 * Abelian-to-normal replacement theorem for prime-fourth order
 * Jonah-Konvisser congruence condition on number of abelian subgroups of small prime power order for odd prime