The maximum possible order of a group \( S_5 \) (the symmetric group on 5 elements) refers to the largest possible order of an element in the group. The order of a group element is the smallest positive integer \( n \) such that the element raised to the power of \( n \) returns the identity element.

To find the maximum order of an element in \( S_5 \), we need to consider the possible cycle types of the elements.

- **Cycle decomposition**: Each permutation in \( S_5 \) can be written as a product of disjoint cycles. The order of a permutation is the least common multiple (LCM) of the lengths of the disjoint cycles in its decomposition.
  
For \( S_5 \), the possible cycle types and their corresponding orders are:

1. **5-cycle** (e.g., \( (1\ 2\ 3\ 4\ 5) \)): The order of this permutation is 5, since it takes 5 steps to return to the identity.
   
2. **4-cycle and 1-cycle** (e.g., \( (1\ 2\ 3\ 4)(5) \)): The order is 4, as the 4-cycle returns to the identity in 4 steps.

3. **3-cycle and 2-cycle** (e.g., \( (1\ 2\ 3)(4\ 5) \)): The order is 6, as the LCM of 3 and 2 is 6.

4. **3-cycle and 1-cycle and 1-cycle** (e.g., \( (1\ 2\ 3)(4)(5) \)): The order is 3, as the 3-cycle has order 3.

5. **2-cycle and 2-cycle and 1-cycle** (e.g., \( (1\ 2)(3\ 4)(5) \)): The order is 2, as the LCM of the two 2-cycles is 2.

Now, to determine the **maximum possible order**, we look for the highest LCM:

- The maximum LCM occurs for a **5-cycle**, which has order 5.

Thus, the **maximum possible order of an element in \( S_5 \) is 5**.

### Summary:
- The maximum order is 5, and it corresponds to a 5-cycle.

Would you like more details on cycle types in symmetric groups or their significance? Here are some follow-up questions:
1. What is the order of a group element in \( S_n \) for other values of \( n \)?
2. How do we find the order of an element in any finite group?
3. What is the structure of the symmetric group \( S_5 \)?
4. How does the order of a permutation relate to its disjoint cycle decomposition?
5. What is the relation between LCM and the order of group elements?

**Tip**: Remember, when calculating the order of a permutation in \( S_n \), always take the LCM of the lengths of its disjoint cycles.

What is the maximum possible order of S5?

Discover the maximum possible order of an element in the symmetric group S5. Explore key group theory concepts, the role of disjoint cycle decompositions, and how to calculate order using the LCM of cycle lengths. Perfect for abstract algebra enthusiasts and students.

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