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 A₆
• B. The alternating group A₅
• C. S₆
• D. S₅

Answer 1

The Correct Option is A

The given problem requires us to identify which group has elements of order 1, 2, 3, 4, 5, but not beyond 5. Let's evaluate each option:

1. The alternating group A₆:
   • The alternating group A₆ is the group of even permutations of a 6-element set. This means A₆ contains only even permutations.
   • The highest order of an element in A₆ is less than 6, as any permutation with order 6 would be derived from an odd permutation (cycle), which A₆ does not include.
   • Thus, A₆ with elements of order up to 5 but no elements of order 6 fits the requirements.
2. The alternating group A₅:
   • A₅ is the group of even permutations of a 5-element set.
   • It does not contain permutations of order 5 or more, as the highest possible order is that of the 3-cycle, which is 3.
   • This does not satisfy the condition as it doesn't include elements of order 4 or 5.
3. The symmetric group S₆:
   • S₆ contains all permutations of a set of 6 elements; hence, it includes permutations of order 6 (like the 6-cycle).
   • This does not fit the requirement as we need a group without elements of order greater than or equal to 6.
4. The symmetric group S₅:
   • S₅ contains permutations of 5 elements and can include orders up to 5 (such as the 5-cycle).
   • However, it does not fit the requirement of having elements of order 4.

Conclusively, The alternating group A₆ has elements of order 1, 2, 3, 4, 5 but no element of order greater than or equal to 6, making it the correct choice.