commutator group: meaning, definition, pronunciation and examples
Very lowTechnical, Academic
Quick answer
What does “commutator group” mean?
In abstract algebra, the subgroup of a given group generated by all commutators (elements of the form g⁻¹h⁻¹gh). It is the smallest normal subgroup such that the quotient group is abelian.
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Pronunciation
Definition
Meaning and Definition
In abstract algebra, the subgroup of a given group generated by all commutators (elements of the form g⁻¹h⁻¹gh). It is the smallest normal subgroup such that the quotient group is abelian.
A measure of how much a group fails to be commutative; the derived subgroup of a group. In some contexts, like the theory of Lie groups, it may refer to the group generated by the commutators in the corresponding Lie algebra.
Dialectal Variation
British vs American Usage
Differences
No lexical or semantic differences. Orthography (e.g., group versus group) shows no variation. The term is equally standard in all mathematical English.
Connotations
Purely technical, no connotative differences.
Frequency
Identically rare and specialized in both varieties. Used only in mathematics textbooks, papers, and lectures.
Grammar
How to Use “commutator group” in a Sentence
The commutator group [of a group G] is often denoted [G, G].Vocabulary
Collocations
Examples
Examples of “commutator group” in a Sentence
verb
British English
- N/A
American English
- N/A
adverb
British English
- N/A
American English
- N/A
adjective
British English
- N/A
American English
- N/A
Usage
Meaning in Context
Business
Never used.
Academic
Exclusive usage. Found in pure mathematics, particularly abstract algebra, group theory, and related fields.
Everyday
Never used. Unintelligible to the general public.
Technical
Core term in advanced mathematics. Used with precise definition.
Vocabulary
Synonyms of “commutator group”
Strong
Neutral
Weak
Vocabulary
Antonyms of “commutator group”
Watch out
Common Mistakes When Using “commutator group”
- Using 'commutator group' to mean a group that is commutative (it means the exact opposite: the part that makes it non-commutative). Misspelling as 'commuter group'. Attempting to use it in non-mathematical contexts.
FAQ
Frequently Asked Questions
No. The center consists of elements that commute with *everything*. The commutator group consists of products of commutators, which measure *failure* to commute. They are different subgroups.
Yes. For example, the commutator group of a non-abelian simple group (like the alternating group A5) is the whole group itself.
Because conjugating a commutator [a,b] by any element g yields another commutator [gag⁻¹, gbg⁻¹]. Thus, the generating set is closed under conjugation, making the whole subgroup normal.
Primarily in abstract algebra (group theory), but it also appears in topological group theory, Lie group/Lie algebra theory, and geometric group theory.
In abstract algebra, the subgroup of a given group generated by all commutators (elements of the form g⁻¹h⁻¹gh). It is the smallest normal subgroup such that the quotient group is abelian.
Commutator group is usually technical, academic in register.
Commutator group: in British English it is pronounced /ˈkɒm.jʊ.teɪ.tə ɡruːp/, and in American English it is pronounced /ˈkɑː.mjə.teɪ.t̬ɚ ɡruːp/. Tap the audio buttons above to hear it.
Phrases
Idioms & Phrases
- “None”
Learning
Memory Aids
Mnemonic
Imagine a GROUP of people who only become friendly (commute their differences) after a formal handshake (the commutator operation). The COMMUTATOR GROUP is the set of all such reconciled friendships needed to make the whole group get along (abelian).
Conceptual Metaphor
THE MEASURE OF DISAGREEMENT. The commutator group is conceptualized as the 'amount of non-commutativity' or the 'obstruction to switching order' within a group.
Practice
Quiz
What is another name for the commutator group?