coset: meaning, definition, pronunciation and examples
Very LowTechnical / Academic
Quick answer
What does “coset” mean?
In mathematics, a set composed of all the products obtained by multiplying each element of a subgroup by a fixed element from the containing group.
Audio
Pronunciation
Definition
Meaning and Definition
In mathematics, a set composed of all the products obtained by multiplying each element of a subgroup by a fixed element from the containing group.
A fundamental concept in abstract algebra (group theory) used to partition a group into equal-sized, non-overlapping subsets relative to a subgroup. It is a building block for defining quotient groups and studying group structure.
Dialectal Variation
British vs American Usage
Differences
No differences in meaning, usage, or spelling. The term is identical in all varieties of English used in mathematics.
Connotations
None beyond its precise mathematical definition.
Frequency
Equally rare and specialised in both dialects, confined to university-level mathematics contexts.
Grammar
How to Use “coset” in a Sentence
[subgroup] + [preposition] + [group] (e.g., 'the coset of H in G')[element] + [subgroup] (e.g., 'the left coset gH')Vocabulary
Collocations
Examples
Examples of “coset” in a Sentence
adjective
British English
- The coset representative was chosen arbitrarily.
- We studied the coset decomposition of the symmetric group.
American English
- The coset representative was chosen arbitrarily.
- We examined the coset structure of the dihedral group.
Usage
Meaning in Context
Business
Not used.
Academic
Exclusively used in advanced mathematics, particularly in abstract algebra, group theory, and related research papers and textbooks.
Everyday
Never used.
Technical
The primary and only context. Used in software for symbolic algebra (e.g., SageMath, GAP) and technical documentation for such systems.
Vocabulary
Synonyms of “coset”
Neutral
Weak
Watch out
Common Mistakes When Using “coset”
- Mispronouncing it as 'co-set' (like 'co-pilot') rather than 'coset' (KOH-set).
- Using it outside of a strict group theory context.
- Confusing left and right cosets when the group operation is not commutative.
- Treating the coset representative as a unique element; any element of a coset can represent it.
FAQ
Frequently Asked Questions
No, it is a strictly technical term in abstract algebra with no application in everyday language, business, or other sciences.
A left coset is of the form gH (multiply the subgroup H on the left by g), while a right coset is Hg (multiply on the right). They are identical for all g if and only if H is a normal subgroup.
Almost never. A coset contains the identity element only if it is the original subgroup itself. Therefore, no other coset has an identity element and cannot be a subgroup or group under the same operation.
Cosets allow mathematicians to partition a group into equal-sized pieces, leading to the powerful concepts of quotient groups, group homomorphisms, and the Fundamental Theorem of Finite Abelian Groups. They are essential for understanding a group's internal structure.
In mathematics, a set composed of all the products obtained by multiplying each element of a subgroup by a fixed element from the containing group.
Coset is usually technical / academic in register.
Coset: in British English it is pronounced /ˈkəʊsɛt/, and in American English it is pronounced /ˈkoʊsɛt/. Tap the audio buttons above to hear it.
Learning
Memory Aids
Mnemonic
Think of a COunter of a SET. A coset is a set you get by starting with a subgroup and 'counting off' from it by a fixed element, creating a parallel set within the larger group.
Conceptual Metaphor
A FAMILY OF TRANSLATED OBJECTS. Imagine a subgroup as a pattern or stamp. A coset is what you get when you 'slide' (translate) that entire pattern to a new starting point within the group, without rotating or distorting it.
Practice
Quiz
What is a necessary and sufficient condition for the set of left cosets to form a group under a natural operation?