Con-Cos group

Definition
A Con-Cos group is a group $$G$$ with a normal subgroup $$N$$ satisfying the following two conditions:


 * Any two non-identity elements of $$N$$ are conjugate in $$G$$.
 * Every non-identity coset of $$N$$ in $$G$$ is a conjugacy class in $$G$$.

Note that if $$G$$ is an abelian group, we can set $$N$$ to be the trivial subgroup.

Stronger properties

 * Weaker than::Abelian group
 * Weaker than::Group with two conjugacy classes

Related properties

 * Camina group
 * Generalized Camina group