Questions tagged [2-groups]

For questions regarding groups of even prime power order, as distinct from p-groups in general. Topics include 2-groups of maximal class, 2-groups as Sylow subgroups, and the conjecture that almost all groups are 2-groups. Not intended for use with the p-groups tag.

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Can we find a non central element of order 2 in a specific 2-group?

Let $G$ be a non-abelian group of order $2^5$ and center $Z(G)$ is non cyclic. Can we always find an element $x\not\in Z(G)$ of order $2$ if for any pair of elements $a$ and $b$ of $Z(G)$ of order $2$,...
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Proportion of nonabelian $2$-groups of a certain order whose exponent is $4$

Let $$\displaystyle A(n)=\frac{\text{number of nonabelian 2-groups of order n whose exponent is }4}{\text{total number of nonabelian 2-groups of order n}}.$$ Using GAP, I could observe the ...
Finite $2$-groups of order $>32$ and nilpotency class $2$
I only know $2$-groups of nilpotency class $2$ and order less than or equal to $32$, and wondering if there are finite $2$-groups of order $>32$ and nilpotency class $2$? Your suggestions are ...