Question

Which one of the following statements is true?

• A. Exactly half of the elements in any even-order subgroup of \(S_5\) must be even permutations.
• B. Any abelian subgroup of \(S_5\) is trivial.
• C. There exists a cyclic subgroup of \(S_5\) of order 6.
• D. There exists a normal subgroup of \(S_5\) of index 7.

Hint:

Disjoint cycles’ orders combine via the LCM rule, so a 3-cycle and a 2-cycle together generate order 6.

Answer 1

The Correct Option is C

Step 1: Analyze cyclic subgroups.
A 6-cycle would have order 6, but \(S_5\) has only 5 symbols. However, a permutation composed of a 3-cycle and a disjoint 2-cycle, such as \((1\,2\,3)(4\,5)\), has order \(\mathrm{lcm}(3,2) = 6\). Hence, it generates a cyclic subgroup of order 6.
Step 2: Check other options.
(A) Not always true. (B) False, since abelian subgroups like \(\langle (1\,2) \rangle\) exist. (D) Index 7 would imply order \(|S_5|/7 = 120/7\), not an integer, impossible.
Step 3: Conclusion.
Hence, only (C) is correct.