Let $(G, \cdot)$ be a group, and $H \leqslant G$. Let $x \in H$, what $C_H(x) < C_G(x)$ mean? ($C_A(x)$ notation for "centralizer of $x$ 'in A")

Question

Let $(G, \cdot)$ be a group. For any $x \in G$, we write:

$$ C_G(x) = \{z \in G \mid z \cdot x = x \cdot z\}$$

Let $H \leqslant G$ (subgroup of), and $x \in H$. What does it mean when we write:

$$ C_H(x) < C_G(x) $$

Does it mean that $\left| C_H(x) \right| < \left| C_G(x) \right|$? Does it mean that $C_H(x) \subset C_G(x)$?

I found this notation in Rotman's Introduction to the Theory of Groups:

Answer 1

It means that $C_H(x)$ is a proper subgroup of $C_G(x)$, i.e. $C_H(x) \subseteq C_G(x)$ and $C_H(x) \neq C_G(x)$.