Why is the generalized quaternion group $Q_n$ not a semi-direct product? Why is the generalized quaternion group $Q_n$ not a semidirect product?
 A: One characterization of the generalized quaternion group is:


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*If $G$ is a non-abelian $p$-group which contains only-one subgroup of order $p$, then $G$ is generalized quaternion group [Hall-Theory of groups; Theorem 12.5.2]. 


So if we try to write the generalized quaternion group $Q_n$ as semi-direct product, then we should have a normal  subgroup $N$, a subgroup $H$, with one necessary condition that $N\cap H=1$; which is not possible because of uniqueness of subgroup of order $p$; it will contained in all subgroups of $Q_n$. Hence the generalized quaternion group is not semi-direct product of smaller $p$-groups.
A: How many elements of order 2 does a generalized quaternion 2-group have?  How many elements of order 2 must each factor in the semi-direct product have?
Note that dicyclic groups (generalized quaternion groups that are not 2-groups) can be semi-direct products.  The dicyclic group of order 24 is a semi-direct product of a group of a quaternion group of order 8 acting on a cyclic group of order 3.
