Is there a group with exactly 92 elements of order 3?
More boldly, I would like to know (but feel free to answer only the first question): Exactly which numbers occur as the number of elements of order 3 in a group? Background: Such questions were studied a bit by Sylow and more heavily by Frobenius. The theorem that the number of elements of order p is equal to −1 mod p is contained in one of Frobenius’s 1903 papers. Since elements of order 3 come in pairs, this doubles to give 2 mod 6 for p=3. However, Frobenius’s results were improved some 30 years later by P. Hall who showed that if the Sylow p-subgroups are not cyclic, then the number of elements of order p is −1 mod p2. If the Sylows are cyclic of order pn, then the number of subgroups of order p is congruent to 1 mod pn by the standard counting method. If the Sylow itself is order p, then the subgroup generated by the elements of order p acts faithfully and transitively on the Sylow subgroups, so for small enough numbers, the subgroup can just be looked up. In all cases, we can ass
