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Is there any example for non-Abelian group in which has more than $3$ elements of order $ 2$, but has less than 7 elements of order $2$?

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The dihedral group of order $8$ - the symmetries of a square is one such.

It has the identity element and two elements of order $4$ (clockwise and counterclockwise quarter turns). The other five elements are four reflections about axes of symmetry and a half turn - all of order 2.

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Consider $D_4$ Dihedral group, non-abelian, $5$ elements of order $2$.

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  • 2
    $\begingroup$ Also dihedral group with 10 elements works. $\endgroup$ – Geoff Robinson Jan 25 '15 at 19:13

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