Transitive and non-transitive subgroups of permutation groups

Arda Demirhan, Jacob Miller, Yixu Qiu, Thomas J. Tucker, Zheng Zhu

Published 2023-11-20Version 1

We treat the problem of finding transitive subgroups G of S_n containing normal subgroups N_1 and N_2, with N_1 transitive and N_2 not transitive, such that G/N_1 is isomorphic G/N_2. We show that such G exist whenever n has a prime factor that also divides the Euler-phi function of n. We show that no such G exist when n = pq for p < q with p not dividing q-1.