Normal of order equal to least prime divisor of group order implies central
This article describes an easy-to-prove fact about basic notions in group theory, that is not very well-known or important in itself
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This page describes additional conditions under which a subgroup property implication can be reversed, viz a weaker subgroup property, namely Normal subgroup (?), can be made to imply a stronger subgroup property, namely central subgroup
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Statement
In a finite group, a normal subgroup whose order is the least prime divisor of the order of the group, must be a central subgroup (viz, it is contianed inside the center).
Related facts
Similar facts
- Normal of order two implies central
- Cyclic normal Sylow subgroup for least prime divisor is central, used to show that cyclic Sylow subgroup for least prime divisor has normal complement
- Minimal normal implies central in nilpotent, leading to minimal normal implies contained in Omega-1 of center for nilpotent p-group or equivalently socle equals Omega-1 of center in nilpotent p-group
Dual facts
Other normal-to-central facts
Proof
Take any element inside the normal subgroup. The size of the conjugacy class of that element is strictly less than the least prime divisor (because the identity element is in a different conjugacy class from other elements). Since the conjugacy class of the element is an orbit of the element under the group action under conjugation, the size of the conjugacy class divides the order of the group. The only divisor of the order of the group, which is less than the least prime divisor, is 1. Thus every conjugacy class has size 1, and the subgroup is thus contained in the center.
