P-central group

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 defining ingredient::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 defining ingredient::center. In the finite case, this is equivalent to saying that the second omega subgroup $$\Omega_2(P)$$ is contained in the center $$Z(P)$$.

Similar properties

 * Powerful p-group