Question

Which one of the following groups has elements of order 1, 2, 3, 4, 5 but does not have an element of order greater than or equal to 6?

• A. The alternating group \( A_6 \)
• B. The alternating group \( A_5 \)
• C. \( S_6 \)
• D. \( S_5 \)

Hint:

In the alternating groups \( A_n \), the order of the elements corresponds to the least common multiple of the lengths of the disjoint cycles in their cycle decomposition. The highest possible order in \( A_6 \) is 5.

Answer 1

The Correct Option is A

The group \( A_n \) is the alternating group on \( n \) elements, consisting of all even permutations of the set \( \{1, 2, ..., n\} \). \( A_6 \) is the alternating group on 6 elements. It contains elements of orders 1, 2, 3, 4, and 5, but does not have elements of order 6 or greater. The largest possible order of an element in \( A_6 \) is 5. \( A_5 \) is the alternating group on 5 elements, and it contains elements of orders 1, 2, 3, and 5, but not 4, so this group does not meet the condition. \( S_6 \) is the symmetric group on 6 elements, and it contains elements of order 6, which violates the condition of not having elements of order 6 or greater. \( S_5 \) is the symmetric group on 5 elements, and it contains elements of order 6 (which is the order of a 5-cycle), which also violates the condition. Therefore, the correct answer is (A) \( A_6 \), as it contains elements of orders 1, 2, 3, 4, and 5, but no elements of order 6 or greater.