this is an exercise in the book "Elementos de Álgebra", a brazillian book of Algebra by Arnaldo Garcia and Yves Lequain (This is not a homework, I'm doing this by myself).
Let $n \geq 3$ an integer. Show that $Q_{n}$ (the generalized quaternion group) is not isomorphic to the dihedral group of order $2^n$, $D_{2^{n-1}}$.
The book itself say that may be usefull to show that $Q_{n}$ has only one element of order 2. This is actually easy and was already solved here. The question, then is reduced to show that there is two (or more) or none element of order 2 in $D_{2^{n-1}}$.
But we know that every reflection on some line of symmetry in $D_{2^{n-1}}$ has order 2, so, as we have $n \geq 3$, there are at least 2 reflections for each $n$.
So, $Q_{n}$ cannot be isomorphic to $D_{2^{n-1}}$; (Isomorphism preserves order).
Am I right? Thanks in advance!