In this article, we characterize finite groups having normal subgroups of a given prime index. Precisely, we prove that if p is a prime divisor of a finite group G, then G has no normal subgroup of index p if and only if G=G’Gp, where Gp is the subgroup of G generated by all elements of the form gp for any g in G and G’ is the derived subgroup of G. We also extend a characterization of finite groups with no subgroups of index 2 by J.B. Nganou to infinite groups. We display an example to show that for a prime index greater than 2 the characterization does not hold.
Olivier A. Heubo-Kwenga
"On the existence of normal subgroups of prime index,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 17:
1, Article 13.
Available at: https://scholar.rose-hulman.edu/rhumj/vol17/iss1/13