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.

Author Bio

Brooklynn Szymoniak a recent graduate of Saginaw Valley State University with a degree in Secondary Education of Math and Biology. She will be teaching high school math at Shepherd High School this fall. Other interests of hers include running, coaching volleyball, and spending time outdoors.