Difference between revisions of "Normal of order equal to least prime divisor of group order implies central"
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Revision as of 01:43, 8 February 2011
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This article describes an easytoprove fact about basic notions in group theory, that is not very wellknown 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
 Cyclic normal Sylow subgroup for least prime divisor is central
 Subgroup of least prime index is normal
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.