Correct option is D
Concept:
Symmetric Group S5 : The symmetric group S5 consists of all permutations of 5 elements.
The order of a permutation in S5 is the least common multiple (LCM) of the lengths of the
disjoint cycles in its cycle decomposition.
Solution:
Cycle Types in S5:
The possible cycle types (in terms of lengths of the disjoint cycles) in S5 are
A 1-cycle (identity element), which has order 1.
A 2-cycle (transposition), which has order 2.
A 3-cycle, which has order 3.
A 4-cycle, which has order 4.
A 5-cycle, which has order 5.
Additionally, we can have products of disjoint cycles, and the order of a product of disjoint cycles is
the least common multiple (LCM) of the lengths of the cycles.
Order 3: This corresponds to a 3-cycle. For example, (1 2 3) has order 3. Hence, 3 is an order of an element in S5.
Order 4: This corresponds to a 4-cycle. For example, (1 2 3 4)has order 4. Hence, 4 is an order of an element in S5.
Order 5: This corresponds to a 5-cycle. For example, (1 2 3 4 5) has order 5. Hence, 5 is an order of an element in S5.
Order 6: A product of a 2-cycle and a 3-cycle has order 6, since the LCM of 2 and 3 is 6. For example, (1 2)(3 4 5) has order 6.
Hence, 6 is an order of an element in S5.
All of the options are correct.
