Centralizer: meaning, definitions and examples

In mathematics and abstract algebra, particularly in group theory, the centralizer of a subset is the set of elements that commute with every element of the subset.

• In group theory, the centralizer of a subset helps to understand the structure of the group.
• The study of centralizers can reveal symmetries within mathematical objects.

The commutator is often used in the context of algebra and physics. It describes a measure of how non-commutative two elements of an algebra are.

• The commutator of two matrices A and B is given by AB - BA.
• Commutators play a crucial role in quantum mechanics and the Heisenberg uncertainty principle.

In group theory, the normalizer of a subgroup is the set of elements that conjugate the subgroup to itself.

• Finding the normalizer of a subgroup can help in understanding its structure within the larger group.
• The normalizer is used to determine whether a subgroup is normal.

The stabilizer in group theory refers to the set of elements that leave a particular element unchanged when acting on it.

• In geometry, the stabilizer of a point in a symmetry group consists of all symmetries that keep the point fixed.
• The concept of a stabilizer is essential in understanding group actions.