How many elements of order two can a nonabelian group have? [duplicate]
Solution 1:
In fact, the threshold to force the group to be elementary abelian is 3/4, see http://arxiv.org/abs/0911.1154 and also https://mathoverflow.net/questions/40028/half-or-more-elements-order-two-implies-generalized-dihedral
Solution 2:
Consider the group of symmetries of a square. $\mathbb{D}_4$. Then, flip along horizontal, vertical and the two diagonals are of order $2$. Also, rotation by 180° is of order $2$