Centralizer as a Subgroup Proof

Show that the centralizer of $H$, defined as $C_H = \{ g \in G \mid gh = hg, \forall h \in H \}$, is a subgroup of $G$.

The centralizer of a subset in group theory is an important concept to understand because it helps us analyze the structure of groups by looking at elements that commute with other elements. When solving problems involving subgroups, recognizing properties such as closure, identity, and inverses, that must be satisfied for a set to be a subgroup, is crucial. Even though proving something is a subgroup might seem repetitive or straightforward, it's essential to meticulously check and confirm these properties.

Knowing that the centralizer is a subgroup of a group has practical implications in areas such as symmetry operations and algebraic geometry. By understanding the subgroup structure, you can gain insight into the symmetry relations within mathematical objects and identify invariant properties under group operations. This proof specifically involves verifying that the set of elements commuting with every element of a subset forms a subgroup, emphasizing the interplay between elements' commutative behaviors and subgroup characteristics.

In this problem, your task is to show that the centralizer of a subset is itself a subgroup by using the group axioms: closure, identity, and inverses. This strengthens your theoretical understanding of group theory and will aid in solving more complex problems that involve analyzing other types of subgroups or group-related structures in mathematics. Always remember, group properties are not just abstract concepts; they provide a framework for exploring the intricate connections within mathematical and physical systems.