Let $G$ be a group of order $2^n$. Does $G$ have a normal abelian subgroup of order at least $2^{n/2}$?


(This is true, via computations in GAP, for $n \le 8$.
The question is similar to one posed here: <https://math.stackexchange.com/questions/44275/abelian-subgroups-of-p-groups/44283#44283>
However, that question, and answer, involves groups of order $p^n$, for odd primes $p$, and I need $p$ to be even!)



Is the largest normal abelian subgroup of a finite 2-group $P$ of order at least the square root of the order of $P$?
In


*Alperin, J. L.*, [**Large abelian subgroups of p-groups**](http://dx.doi.org/10.2307/1994193), Trans. Am. Math. Soc. 117, 10-20 (1965). [ZBL0132.27204](https://zbmath.org/?q=an:0132.27204),


the second part of Theorem 1 gives a group of order $2^{50}$ with no abelian subgroups of order greater than $2^{24}$.


