Question

The order of the element \( (1 \ 2 \ 3)(2 \ 4 \ 5)(4 \ 5 \ 6) \) in the group \( S_6 \) is ..............

Hint:

The order of a permutation is the least common multiple (LCM) of the lengths of its disjoint cycles.

Answer 1

Correct Answer: 4

1. Simplify the Permutation

The given permutation is not in disjoint cycle form, so the first step is to compose the cycles to find the disjoint cycle decomposition:

$$\alpha = (1 \ 2 \ 3)(2 \ 4 \ 5)(4 \ 5 \ 6)$$

We track the movement of each element from 1 to 6, working from right to left:

1 $\xrightarrow{\text{3rd cycle}}$ 1 $\xrightarrow{\text{2nd cycle}}$ 1 $\xrightarrow{\text{1st cycle}}$ 2

2 $\xrightarrow{\text{3rd cycle}}$ 2 $\xrightarrow{\text{2nd cycle}}$ 4 $\xrightarrow{\text{1st cycle}}$ 4

4 $\xrightarrow{\text{3rd cycle}}$ 5 $\xrightarrow{\text{2nd cycle}}$ 2 $\xrightarrow{\text{1st cycle}}$ 3

3 $\xrightarrow{\text{3rd cycle}}$ 3 $\xrightarrow{\text{2nd cycle}}$ 3 $\xrightarrow{\text{1st cycle}}$ 1 (Cycle closes: $(1 \ 2 \ 4 \ 3)$)

5 $\xrightarrow{\text{3rd cycle}}$ 6 $\xrightarrow{\text{2nd cycle}}$ 6 $\xrightarrow{\text{1st cycle}}$ 6

6 $\xrightarrow{\text{3rd cycle}}$ 4 $\xrightarrow{\text{2nd cycle}}$ 5 $\xrightarrow{\text{1st cycle}}$ 5 (Cycle closes: $(5 \ 6)$)

The disjoint cycle decomposition is:

$$\alpha = (1 \ 2 \ 4 \ 3) (5 \ 6)$$

The cycle lengths are $\mathbf{4}$ and $\mathbf{2}$.

2. Determine the Order

The order of a permutation is the Least Common Multiple (LCM) of the lengths of its disjoint cycles.

$$\text{Order}(\alpha) = \text{lcm}(4, 2) = \mathbf{4}$$