Tarski monster group

An infinite group whose every proper nontrivial subgroup is cyclic of a fixed prime order $p$. Such groups exist for sufficiently large primes. This subgroup condition is stronger than the condition that every nonidentity element has order $p$.

// Target: algebra.bigb

For a Tarski monster group, any nonidentity element generates a proper subgroup of order $p$. Adjoining an element outside that subgroup generates the whole group. A proper nontrivial normal subgroup would have order $p$; conjugation into its finite automorphism group forces it to be central. Combining it with another order-$p$ cyclic subgroup would then give a proper subgroup of order $p^2$, a contradiction.

// Target: algebra.bigb