← 2013 Paper 2

UPSC 2013 Maths Optional Paper 2 Q2a — Step-by-Step Solution

10 marks · Section A

Permutation Groups (S_n): Cycle Decomposition, Sign, A_n · Algebra · asked 2× in 13 yrs · Read the full method →

Question

What are the orders of the following permutations in $S_{10}$?

Technique

Disjoint-cycle decomposition; lcm of cycle lengths is the order.

Solution

Strategy. The order of a permutation equals the lcm of its disjoint-cycle lengths.

Permutation 1

Trace out the cycles by following $1\to\sigma(1)\to\sigma^{2}(1)\to\ldots$:

• $1\to 1$ — fixed point (1-cycle).
• $2\to 8\to 2$ — 2-cycle $(2\,8)$.
• $3\to 7\to 4\to 3$ — 3-cycle $(3\,7\,4)$.
• $5\to 10\to 9\to 6\to 5$ — 4-cycle $(5\,10\,9\,6)$.

Cycle decomposition: $\sigma=(2\,8)(3\,7\,4)(5\,10\,9\,6)$ with one fixed point.

Permutation 2

$\tau=(1\,2\,3\,4\,5)(6\,7)$ — a 5-cycle and a 2-cycle, already in disjoint form.

Final answer

Answer