P-central group

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Definition

For the case of an odd prime

Let p be an odd prime. A p-central group is a p-group (i.e., a group in which the order of every element is a power of a fixed prime number p) with the property that all the elements of order p are inside the center. In the finite case, this is equivalent to saying that the first omega subgroup \Omega_1(P) is contained in the center Z(P).

For the case p = 2

For the prime p = 2, a p-central group (or a 2-central group in this case) is a p-group (i.e., a group in which the order of every element is a power of a fixed prime number p) with the property that all elements of order p or p^2 are in the center. In the finite case, this is equivalent to saying that the second omega subgroup \Omega_2(P) is contained in the center Z(P).

Relation with other properties

Similar properties