Let $H \leqslant G$ be abelian groups. Suppose there were $k \gt 1$, $x \in G \setminus H$, such that $x^k = b \in H$. Then Define $H(x) = \{h x^n : n \in \Bbb{Z}, h \in H\}$ to be a simple extension of $G$ obtained by adjoining $x$. It is such that $H \leqslant H(x) \leqslant G$, and also $H(x)$ is an algebraic extension as each $h' \in H(x)$ is the root of some $x^m = b \in H$. There's probably a lot more to be said.
Has anyone looked into algebraic extensions of just groups (not fields) and not the usual definition of group extension (having to do with exact sequences), but the one above, ie. "algebraic".
Thanks.