← 2013 Paper 2

UPSC 2013 Maths Optional Paper 2 Q2b — Step-by-Step Solution

13 marks · Section A

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

Question

What is the maximal possible order of an element in $S_{10}$? Why? Give an example of such an element. How many elements will there be in $S_{10}$ of that order?

Technique

Landau’s function $g(n)$ — maximum lcm of partitions of $n$; counting formula for permutations by cycle type.

Solution

Strategy. Order of an element in $S_n$ = lcm of its cycle lengths. The maximum over $S_{10}$ corresponds to the partition of 10 with maximum lcm of parts (sometimes called Landau’s function $g(10)$).

Step 1 — Find the partition of 10 with maximum lcm

Search over partitions of 10 (only the lcm-maximising ones listed):

(Smaller-lcm partitions omitted.)

The maximum lcm is $\boxed{30}$, achieved uniquely by the partition $5+3+2$.

Step 2 — Why? (Brief reason)

To maximise $\operatorname{lcm}$ of parts summing to 10, we want parts that are coprime (or share few factors). $\{5,3,2\}$ are pairwise coprime, so lcm = product = 30. No other partition of 10 reaches this product. (Partitions like $7+3$ give lcm $21<30$; $5+4+1$ gives $20<30$.)

Step 3 — Example of an order-30 element

Any product of disjoint cycles with lengths $5,\,3,\,2$ works. For instance:

Order $=\operatorname{lcm}(5,3,2)=30$ ✓.

Step 4 — Count of order-30 elements in $S_{10}$

An order-30 element has cycle type $5+3+2$ (uniquely). The number of permutations of a given cycle type $(k_1^{m_1},k_2^{m_2},\ldots)$ in $S_n$ is

Here cycle type is one 5-cycle, one 3-cycle, one 2-cycle — multiplicities all 1:

Answer