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Centralizer Meaning: Definition, Examples, and Translations

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centralizer

[ˈsɛntrəˌlaɪzər ]

Definition

Context #1 | Noun

mathematics

A centralizer is a subgroup of a group that commutes with every element of the group.

Synonyms

commutator, normalizer, stabilizer.

Which Synonym Should You Choose?

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Word Description / Examples
centralizer

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.
commutator

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.
normalizer

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.
stabilizer

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.

Examples of usage

  • The centralizer of an element in a group is the set of elements that commute with that element.
  • The centralizer of the identity element in a group is the whole group itself.

Translations

To see the translation, please select a language from the options available.

Interesting Facts

Technology

  • In computing, a centralizer can refer to software that consolidates data to improve access and management.
  • Cloud computing often uses centralizers to gather various services into a single platform, making it easier for users to interact with multiple tools.

Business

  • Centralizers in business refer to organizational strategies that focus decision-making power in a central location or authority.
  • Companies might use centralization as a strategy to improve efficiency by coordinating various departments under a unified leadership.

Education

  • In education systems, centralizers can refer to methodologies or curricula that unify various subjects to present a cohesive learning experience.
  • Centralizing educational resources can help students access information more easily, leading to improved learning outcomes.

Cultural Sociology

  • Centrally organized events, such as festivals or community gatherings, can serve as 'centralizers' by bringing diverse groups together.
  • Centralization in social networks is often analyzed for its impact on communal ties, showing how platforms can either connect or isolate users.

Origin of 'centralizer'

Main points about word origin

  • The word comes from 'central,' meaning the middle point, and 'izer,' which indicates something that makes it central.
  • The use of '-izer' formations in English often signifies an agent or entity that performs an action.

The term 'centralizer' in mathematics originated from the concept of centralizing elements in a group. It has applications in group theory, algebra, and other branches of mathematics.

See also: centralise, centralize, centralized, centralizing, decentralization.