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Dear all, Does exist 4 distinct elements of order 4 in extra-special 2-groups? yours,

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An extraspecial 2-group is isomorphic to a central product of copies of the dihedral group $D_8$ and the quaternion group $Q_8$, both of order 8. Since the central product of two $D_8$s is isomorphic to that of two $Q_8$s, there are just two isomorphism types of extraspecial 2-groups of each order $2^{1+2n}$.

Your statement is false in $D_8$, which has only 2 elements of order 4. All other extraspecial 2-groups contain $Q_8$ as a subgroup, which has 6 distinct elements of order 4.

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Incidentally, this is how to prove that for each $n,$ there are two different isomorphism types of extra-special groups of order $2^{2n+1}$. One is a central product of $D_{8}$ with $n-1$ copies quaternion groups of order $8.$ The other is a central product of $n$ quatrnion groups of order $8.$ These groups have different numbers of elements of order $4$ so they certainly aren't isomorphic. –  Geoff Robinson Jul 21 '12 at 19:35

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