Transposition

Definition

Let $X$ be a set and let $a, b \in X$ with $a \neq b$. The transposition $(a\ b)$ is the permutation of $X$ that sends $a$ to $b$, sends $b$ to $a$, and fixes each other element of $X$.

Remarks

A transposition is a $2$-cycle in cycle notation, and $(a\ b) = (b\ a)$. Each transposition is its own inverse, so it has order $2$. The symmetric group $S_n$ contains exactly $\binom{n}{2}$ transpositions.

Explanations

L1 human

A transposition swaps two things and leaves everything else in place. If you do the same swap two times, you are back at the start. All permutations can be built from transpositions.