A subgroup satisfying the condition $g \not\in H \Rightarrow H \cap g^{-1}Hg=1$ is called *malnormal*. This class of subgroups has been much studied.

For many types of infinite groups, such as word-hyperbolic groups, there are lots of examples. As a simple example, if $G$ is free and $a$ is an element in a free generating set, then $H = \langle a \rangle$ is malnormal in $G$.

If $b$ is another element in the free basis, then neither $b$ nor $b^{-1}a$ lies in a conjugate of $H$, but their product does, so there is no equivalent of the Frobenius kernel.