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Why is the generalized quaternion group $Q_n$ not a semidirect product?

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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.

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    @Babak: Exactly! The factors cannot share any elements of order 2, but they each must have one. There is no room for a semi-direct product.2011-03-15
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One characterization of the generalized quaternion group is:

  • 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.

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    Thanks Rahul for the answer.2011-03-16