alternating group: meaning, definition, pronunciation and examples
C1Academic / Technical / Scientific
Quick answer
What does “alternating group” mean?
A specific mathematical structure.
Audio
Pronunciation
Definition
Meaning and Definition
A specific mathematical structure; in group theory, the subgroup of a symmetric group consisting of all even permutations.
A fundamental concept in abstract algebra and group theory, often denoted Aₙ, where n is the degree. It plays a crucial role in the classification of finite simple groups and has applications in geometry and cryptography.
Dialectal Variation
British vs American Usage
Differences
No difference in meaning or usage. Spelling conventions for surrounding text (e.g., 'behaviour' vs. 'behavior', 'centre' vs. 'center') may apply, but the term itself is invariant.
Connotations
None beyond its strict mathematical definition.
Frequency
Used with identical, very low frequency in both UK and US academic mathematical discourse.
Grammar
How to Use “alternating group” in a Sentence
The alternating group on n letters is...Aₙ is the alternating group of degree n.The group A₅ is a simple alternating group.Vocabulary
Collocations
Examples
Examples of “alternating group” in a Sentence
adjective
British English
- The alternating group structure is central to the proof.
- We focused on alternating group properties.
American English
- The alternating group structure is central to the proof.
- We focused on alternating group properties.
Usage
Meaning in Context
Business
Not used.
Academic
Core term in advanced undergraduate and postgraduate mathematics, particularly in courses on abstract algebra and group theory.
Everyday
Not used.
Technical
Used in research papers, textbooks, and lectures on pure mathematics and related fields like cryptography.
Vocabulary
Synonyms of “alternating group”
Neutral
Weak
Vocabulary
Antonyms of “alternating group”
Watch out
Common Mistakes When Using “alternating group”
- Using 'alternating' to mean 'changing' or 'rotating' in this context. Its meaning is strictly tied to 'even permutations'.
- Misspelling as 'alternative group'.
- Attempting to use it in non-mathematical contexts.
FAQ
Frequently Asked Questions
No. The symmetric group Sₙ contains all permutations of n objects. The alternating group Aₙ is a specific subgroup of Sₙ containing only the even permutations. It has exactly half the number of elements of Sₙ for n > 1.
The name originates from the alternating polynomial, a function whose sign changes under the transposition (swap) of two variables. The permutations that leave such a polynomial invariant (up to sign) are precisely the even permutations, which form the alternating group.
You would encounter it almost exclusively in university-level mathematics courses or textbooks on abstract algebra, group theory, or Galois theory. It is not a term used in general English.
Yes, A₄ is the alternating group of degree 4. It consists of the 12 even permutations of four objects. It is noteworthy for being the smallest alternating group that is not simple, as it contains a normal subgroup.
A specific mathematical structure.
Alternating group is usually academic / technical / scientific in register.
Alternating group: in British English it is pronounced /ˈɔːltəneɪtɪŋ ɡruːp/, and in American English it is pronounced /ˈɔːltərneɪtɪŋ ɡruːp/. Tap the audio buttons above to hear it.
Learning
Memory Aids
Mnemonic
Think of ALTERNATING like switching places: In an 'alternating group', you only count the permutations you can make by swapping an EVEN number of objects back and forth.
Conceptual Metaphor
A highly selective club (the alternating group) within a larger society (the symmetric group), where membership requires a specific, even-handed nature (even parity).
Practice
Quiz
What is the alternating group Aₙ?