# P-central group

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 $Z(P)$.