The class of groups which have a subgroup of index 2 is not elementary

F. Oger proved that if A is a nite group, then the class of groups which are abelian-by-A can be axiomatized by a single rst order sentence. It is established here that, in Oger's result, the word abelian cannot be replaced by group. In 2] it was proved that, if A is a nite group, then the class of groups G which have an abelian subgroup H such that G=H is isomorphic to A can be axiomatized by a single rst order sentence. Professor G. Sabbagh suggested that in this result the word abelian cannot be deleted. This, and more, is established in the present note. We consider exclusively the case where A is the group with two elements, hence the title of this note. For any group G and any integer n we denote by G n the subgroup generated by the n th-powers of elements of G. We denote by C the class of groups which have a subgroup of index 2. Theorem 1 The classe C is not elementary. More precisely C is not closed under elementary substructures. It is clear that C is closed under ultraproducts. The rst step of the proof of theorem 1 is the following very simple charac-terisation of C.