P-constrained group

Definition for a general finite group
Let $$G$$ be a finite group and $$p$$ be a prime number. We say that $$G$$ is $$p$$-constrained if the following is true for one (and hence, any) $$p$$-Sylow subgroup of $$G$$:

$$C_G(P \cap O_{p',p}(G)) \le O_{p',p}(G)$$.

Here, $$C_G(P)$$ denotes the defining ingredient::centralizer of $$P$$ in $$G$$. $$O_{p',p}$$ is the second member of the defining ingredient::lower pi-series for $$\pi = \{ p \}$$.

Definition for a p'-core-free finite group
This is the same as the previous definition, restricted to p'-core-free groups.

Let $$G$$ be a finite group and $$p$$ be a prime number. Suppose further that the p'-core of $$G$$ is trivial, i.e., $$O_{p'}(G)$$ is the trivial group. Equivalently, every nontrivial normal subgroup of $$G$$ has order divisible by $$p$$. Then, we say that $$G$$ is $$p$$-constrained if its defining ingredient::p-core is a defining ingredient::self-centralizing subgroup, i.e.,:

$$\! C_G(O_p(G)) \le O_p(G)$$

Equivalence of definitions and its significance
It turns out that, from the above definitions:

$$G$$ is $$p$$-constrained $$\iff$$ $$G/O_{p'}(G)$$ is $$p$$-constrained.

This allows us to define $$p$$-constraint for arbitrary finite groups in terms of $$p$$-constraint for $$p'$$-core-free finite groups.