Cyclic normal Sylow subgroup for least prime divisor is central

From Groupprops
Jump to: navigation, search
This page describes additional conditions under which a subgroup property implication can be reversed, viz a weaker subgroup property, namely Central subgroup (?), can be made to imply a stronger subgroup property, namely normal subgroup
View other subgroup property implication-reversing conditions
This result relates to the least prime divisor of the order of a group. View more such results


Suppose p is the least prime divisor of the order of a finite group G. Suppose S is a p-Sylow subgroup that is also a cyclic normal subgroup. In other words, S is a normal Sylow subgroup that is cyclic as a group. Then, S is a central subgroup of G.

Related facts

Other facts about least prime divisor