The "generalized dihedral group" for an abelian group *A* is the semidirect product of *A* and a cyclic group of order two acting via the inverse map on *A*. *A* thus has index two in the whole group and all elements outside *A* have order two. Thus, at least half the elements of any generalized dihedral group have order two.


My question is the converse: if half or more the elements of a finite group *G* have order two, is it necessary that *G* is either an elementary abelian 2-group or a generalized dihedral group? [Note: Actually nontrivial elementary abelian 2-groups are also generalized dihedral, they're an extreme case. Also, note that the direct product of a generalized dihedral group with an elementary abelian 2-group is still generalized dihedral, because the elementary abelian 2-group can be pulled inside the abelian part.]


I have a proof when the order of *G* is twice an odd number *m*. In that case, there is a short an elegant elementary proof -- we consider the set of elements that can be written as a product of two elements of order two and show that this is a subgroup of order *m*, then show that any element of order two acts by the inverse map on it, and hence the subgroup is abelian. It can also be thought of as a toy example of Frobenius' theorem on Frobenius subgroups and complements (though we don't need Frobenius' theorem).


However, I am having some difficulty generalizing this, mainly because there are elements of order two that are inside the abelian group. 


Although I have a number of possible proof pathways, I'll refrain from mentioning them for now because the actual proof is likely to be much simpler and I don't want to bias others trying the problem.



Half or more elements order two implies generalized dihedral?
$D\_8\times D\_8$ is such a group (it has 35 involutions) that fails to be either elementary abelian or generalized dihedral.


There is an actual classification of groups with at least half the elements being involutions; it was first done by Miller. A modern reference would be the paper by Wall, ["On Groups Consisting Mostly of Involutions"](https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/abs/on-groups-consisting-mostly-of-involutions/554A7AE2A5C7B537B76CB098A8C6E2E2). A simpler treatment, though with a different goal in mind, is given in Liebeck and MacHale's paper ["Groups with Automorphisms Inverting Most Elements"](https://doi.org/10.1007/BF01142582).


