To determine the maximum possible order of an element in $S_n$, we need to analyze the cycle structure of elements in $S_n$. The order of an element in $S_n$ is given by the least common multiple (LCM) of the lengths of the disjoint cycles in its cycle decomposition.

For $n = 5$, the maximum order will correspond to the LCM of cycle lengths that sum up to $n = 5$. Let’s consider the possibilities:

Step 1: Possible Cycle Decompositions

The partitions of $5$ (the ways of splitting $n = 5$) are:

1. $5$ (one 5-cycle)
2. $4 + 1$ (a 4-cycle and a 1-cycle)
3. $3 + 2$ (a 3-cycle and a 2-cycle)
4. $3 + 1 + 1$ (a 3-cycle and two 1-cycles)
5. $2 + 2 + 1$ (two 2-cycles and one 1-cycle)
6. $2 + 1 + 1 + 1$ (a 2-cycle and three 1-cycles)
7. $1 + 1 + 1 + 1 + 1$ (five 1-cycles)

Step 2: Compute LCM of Cycle Lengths

For each case:

1. $5$: $\text{lcm}(5) = 5$
2. $4 + 1$: $\text{lcm}(4, 1) = 4$
3. $3 + 2$: $\text{lcm}(3, 2) = 6$
4. $3 + 1 + 1$: $\text{lcm}(3, 1, 1) = 3$
5. $2 + 2 + 1$: $\text{lcm}(2, 2, 1) = 2$
6. $2 + 1 + 1 + 1$: $\text{lcm}(2, 1, 1, 1) = 2$
7. $1 + 1 + 1 + 1 + 1$: $\text{lcm}(1, 1, 1, 1, 1) = 1$

Step 3: Maximum Order

The maximum LCM occurs in case $3 + 2$, giving: $\text{Maximum order} = 6$

Conclusion:

The maximum possible order of an element in $S_5$ is 6, achieved by an element with a cycle structure of $(3)(2)$.

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Related Questions:

1. What is the general formula to determine the maximum order of elements in $S_n$?
2. How is the LCM used in finding the order of a permutation in $S_n$?
3. Can the cycle structure $4 + 1$ achieve the maximum order for $n = 5$?
4. What is the significance of partitions in understanding $S_n$?
5. How does the order of elements in $S_n$ relate to group theory concepts like subgroup structure?

Tip:

To compute the order of an element in $S_n$, always focus on finding the least common multiple of cycle lengths in its cycle decomposition.