19.80 (2018)

For a periodic group $G$, let $\pi_e(G)$ denote the set of orders of elements of $G$. A periodic group $G$ is said to be an $OC_n$-group if $\pi_e(G) = \{1, 2, \dots, n\}$. Is it true that every $OC_7$-group is isomorphic to the alternating group $A_7$? For finite groups, the answer is affirmative.