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Let $G$ be a finite group that is neither cyclic, nor a direct product of cyclic groups. Suppose $g \in G$ is an element of order $\frac{|G|}{2}$. Then what could be said about the structure of $G$?

Dihedral groups are examples of this class of groups and I was wondering if more structural properties could be determined based on this property.

Thank you for your insights!

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    $\begingroup$ $g$ generates a subgroup of index $2$, so that's a place to start. Also the Quaternion group is another example of such a group. $\endgroup$ – Josh B. Aug 21 '18 at 0:53
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    $\begingroup$ You should be able to classify such groups. The odd Sylow subgroups are cyclic and normal, the Sylow $2$-subgroups has a cyclic subgroup of index $2$, and those groups are classified (groupprops.subwiki.org/wiki/…). The group will be a semidirect product of its characteristic group of odd order and, a Sylow 2-subgroup. The Sylow $2$-subgroup acts on the other Sylows as an element of order at most $2$, so either trivially or by inversion. $\endgroup$ – verret Aug 21 '18 at 1:20
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$G$ must be of the following form: $G=C\times (D \rtimes P_2)$, where $C$ and $D$ are cyclic of odd coprime order, $P_2$ is a $2$-group having a cyclic subgroup of index $2$, say $C_2$, and $P_2$ acts on $D$ by inversion, with kernel $C_2$.

(In other words, every element in $C_2$ centralises $D$, whereas every element in $P_2$ outside $C_2$ acts by inversion on $D$.)

Note that the possibilities for $P_2$ are well-known (see comments for example). In particular, for a given order, there are at most $6$ isomorphisms types.

Note that every $G$ of this form has the property you want, so this is a complete characterisation. The proof goes roughly as in the comments.

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