even permutation
C2Technical (mathematics, computer science, physics)
Definition
Meaning
A rearrangement of a set of objects that can be achieved by an even number of pairwise swaps (transpositions).
In group theory (mathematics), an even permutation is an element of the alternating group, which forms a subgroup of the symmetric group of all permutations. It is a fundamental concept in abstract algebra, combinatorics, and has applications in physics (e.g., fermionic wave functions).
Linguistics
Semantic Notes
The concept is defined only for finite sets. The parity (evenness or oddness) is invariant; a permutation cannot be both. 'Even' here is unrelated to the number's property of divisibility by two, but refers to the parity of the permutation's decomposition.
Dialectal Variation
British vs American Usage
Differences
No significant lexical or conceptual differences. Spelling of related terms may differ (e.g., 'factorise' vs. 'factorize').
Connotations
Purely technical, with no cultural connotations.
Frequency
Used with identical frequency and meaning in academic and technical contexts in both regions.
Vocabulary
Collocations
Grammar
Valency Patterns
The permutation [X] is an even permutation.[Subject] decomposes into an even permutation.The product results in an even permutation.Vocabulary
Synonyms
Neutral
Vocabulary
Antonyms
Usage
Context Usage
Business
Not used.
Academic
Central concept in undergraduate courses on abstract algebra, group theory, and linear algebra (determinants).
Everyday
Not used.
Technical
Used in algorithm design (sorting networks), cryptography, quantum computing (for parity considerations), and theoretical physics.
Examples
By Part of Speech
adjective
British English
- The permutation's even nature simplifies the calculation.
- We are only interested in even permutation groups.
American English
- The even permutation property is key to the proof.
- An even permutation matrix has a determinant of 1.
Examples
By CEFR Level
- The concept of an even permutation is introduced in advanced mathematics courses.
- Swapping two items creates an odd permutation, not an even one.
- The theorem states that every even permutation can be expressed as a product of 3-cycles.
- Determining whether a given shuffle of a deck is an even permutation requires analysing its cycle structure.
Learning
Memory Aids
Mnemonic
Think of a square dance where couples swap partners. If it takes an EVEN number of two-person swaps to get from the starting lineup to the final lineup, that's an EVEN permutation.
Conceptual Metaphor
A rearrangement with balanced, pairwise trades (like a fair, reciprocal exchange requiring an even number of steps).
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Avoid literal translation of 'even' as 'чётный' in a numerical sense out of context. The term is a fixed compound: 'чётная перестановка'.
- Do not confuse with 'permutation' as a general rearrangement; the 'even/odd' classification is a specific, advanced property.
Common Mistakes
- Using 'even permutation' to describe a permutation that results in an ordered sequence.
- Confusing it with a permutation consisting only of even numbers.
- Assuming the identity permutation is odd (it is even, as it corresponds to zero swaps).
Practice
Quiz
What is the sign (or parity) of an even permutation?
FAQ
Frequently Asked Questions
The identity permutation is an even permutation, as it can be considered as being achieved by zero transpositions (an even number).
They are relevant in computer science algorithms (like the 15-puzzle solvability), in physics regarding the symmetry of wave functions for identical particles, and in the study of Rubik's Cube configurations.
You can decompose it into transpositions (pairwise swaps) and count them—if the count is even, it's an even permutation. More efficiently, you can write it in cycle notation; it is even if the number of cycles of even length is even (for a permutation on a finite set).
It is called the alternating group on n symbols, denoted A_n.