odd permutation
C2Technical/Academic
Definition
Meaning
In mathematics, a permutation that can be expressed as the product of an odd number of transpositions (pairwise swaps).
A concept in group theory and combinatorics describing a specific type of rearrangement of a set's elements, characterized by having a negative sign or determinant of -1 when represented as a matrix.
Linguistics
Semantic Notes
The term is strictly mathematical; 'odd' refers to parity (odd vs. even), not to strangeness. It is defined in opposition to 'even permutation'.
Dialectal Variation
British vs American Usage
Differences
No significant lexical or definitional differences. Spelling of related terms may follow regional conventions (e.g., 'factorise' vs. 'factorize').
Connotations
Identical technical connotations in both varieties.
Frequency
Used with identical frequency in academic mathematics contexts in both regions.
Vocabulary
Collocations
Grammar
Valency Patterns
The permutation [NP] is an odd permutation.An odd permutation [VP] has a sign of -1.To determine if [NP] is an odd permutation, [VP].Vocabulary
Synonyms
Neutral
Vocabulary
Antonyms
Usage
Context Usage
Business
Not used.
Academic
Used in university-level mathematics, particularly in courses on abstract algebra, group theory, and linear algebra.
Everyday
Not used.
Technical
Used in pure mathematics, cryptography (related to permutation groups), and some areas of computer science theory.
Examples
By Part of Speech
adjective
British English
- The resulting permutation was odd.
- We need to identify the odd permutations in the set.
American English
- The resulting permutation was odd.
- We must identify the odd permutations in the set.
Examples
By CEFR Level
- In basic combinatorics, a permutation can be classified as odd or even.
- Swapping two elements creates an odd permutation.
- The determinant of a matrix representing an odd permutation is -1.
- Every odd permutation can be decomposed into a product of an odd number of transpositions.
- The alternating group A_n consists solely of even permutations, excluding all odd permutations.
Learning
Memory Aids
Mnemonic
Think of swapping two items as one 'swap'. An ODD permutation needs an ODD number of such swaps to build it.
Conceptual Metaphor
PARITY IS SIGN (An odd permutation has a 'negative' signature).
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Avoid translating 'odd' as 'странный'. The correct mathematical term is 'нечётная перестановка'.
- Do not confuse with 'random permutation' ('случайная перестановка').
Common Mistakes
- Using 'odd' in its everyday sense (strange).
- Confusing it with 'random permutation'.
- Incorrectly calculating the sign/parity of a permutation.
Practice
Quiz
What is the sign of an odd permutation?
FAQ
Frequently Asked Questions
No. 'Odd' refers to the mathematical parity (odd/even) of the permutation, not to its probability or strangeness. A random permutation could be either odd or even.
Yes. For the set {1,2,3}, the permutation that swaps 1 and 2 (resulting in 2,1,3) is an odd permutation, as it is a single transposition.
It is fundamental in group theory, defining important structures like the alternating group. It is also crucial in calculating determinants in linear algebra and has applications in puzzle theory (e.g., the 15-puzzle) and cryptography.
One common method is to decompose the permutation into a product of transpositions (swaps). If the number of transpositions is odd, the permutation is odd. Alternatively, calculate its sign or parity via inversion count or matrix representation.