P-central group

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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

• Powerful p-group