Definition:Transposition
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Definition
A transposition (colloquially known as a two-letter swap) is a 2-cycle.
That is, a transposition is a permutation $\rho$ on a set $S$ which exchanges, or transposes, exactly two elements of $S$.
Thus if $\rho$ is a transposition which transposes two elements $r, s \in S$, it follows from the definition of fixed elements that:
- $\operatorname{Fix} \left({\rho}\right) = S \setminus \left\{{r, s}\right\}$
Sources
- Ian D. Macdonald: The Theory of Groups (1968): Appendix
- Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 79$
- John F. Humphreys: A Course in Group Theory (1996): $\S 9$
