Let $G$ be an uncountable universal locally finite group. We study subgroups $H<G$ such that for every $g\in G$, $|H:H\cap H^g|<|H|$.

In classifying certain infinite groups under minimal conditions it is needed to find non-simplicity criteria for the groups under consideration. We obtain some of such criteria as a consequence of the main result of the paper and the classification of finite simple groups.

In this paper we deal with the class ${\mathcal{E}}_k^{*}$ of groups $G$ for which whenever we choose two infinite subsets $X$, $Y$ there exist two elements $x\in X$, $y\in Y$ such that $\left[x,\underset{k}{\underbrace{y,\mathrm{\dots },y}}\right]=1$. We prove that an infinite finitely generated soluble group in the class ${\mathcal{E}}_k^{*}$ is in the class ${\mathcal{E}}_k$ of $k$-Engel groups. Furthermore, with $k=2$, we show that if $G\in {\mathcal{E}}_2^{*}$ is infinite locally soluble or hyperabelian group then $G\in {\mathcal{E}}_2$.