Permutations
Permutations: composition, cycle decomposition, the order — how many times to repeat until everything returns — parity and the inverse.
How to solve it
A permutation tells where each number goes: the top row is where from, the bottom row where to. Almost everything about a permutation shows once it is split into cycles — chains like $1 \to 4 \to 1$ and $2 \to 6 \to 5 \to 2$: the order, the parity, the inverse.
Step by step
- Cycles: start at $1$ and follow the arrows until you come back; then take the smallest number not yet visited.
- The composition $\sigma \circ \tau$: first $\tau$, then $\sigma$. For each $i$ find $\tau(i)$, then $\sigma(\tau(i))$.
- The order is the least common multiple of the cycle lengths.
- Parity: a cycle of length $k$ is $k - 1$ transpositions. Add them up and look at the parity of the total.
- The inverse: reverse the arrows — swap the rows and sort the top one.
Common mistakes
- Composing left to right: in $\sigma \circ \tau$ the permutation $\tau$ acts first.
- Adding the cycle lengths instead of taking the lcm: cycles of lengths $2$ and $3$ give the order $6$, not $5$.
- Counting a cycle of length $k$ as $k$ transpositions — it is $k - 1$.
- Leaving out the fixed points when counting cycles for the sign.