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 ?

1. The alternating group $A_6$
2. The alternating group $A_5$
3. $S_6$
4. $S_5$

Answer: The alternating group $A_6$

Solution :-

Option (1)

This option is right because $A_6$ has elements of order 1, 2, 3, 4, 5 and those are listed below.

• elements with order 1 is identity and this is always even.
• elements of order 2 with cycle decomposition <1, 1, 2, 2> is even permutation so all elements with this cycle decomposition belonging in $A_6$ .
• elements of order 3 with cycle decomposition <3, 3> is even permutation so all elements with this cycle decomposition belonging in $A_6$ .
• elements of order 4 with cycle decomposition <2, 4> is even permutation so all elements with this cycle decomposition belonging in $A_6$ .
• elements of order 5 with cycle decomposition <1, 5> is even permutation so all elements with this cycle decomposition belonging in $A_6$ .

But this group has no elements whose order is greater than or equal to 6. Because this group has no even permutation whose order is greater than or equal to 6.

Option (2)

This option is wrong because $A_5$ does not have elements of order 4.
All elements whose order is 4 in $S_5$ are those whose cycle decomposition is <1, 4> but all permutations with this cycle decomposition are odd. So $A_5$ does not have elements of order 4.

Option (3)

This option is also wrong because $S_6$ has an element of order 6 whose cycle decomposition is < 6 >.

Option (4)

This option is also wrong because $S_5$ has an element of order 6 whose cycle decomposition is < 2, 3 >.

So, the correct answer is option-1, $\to$ The alternating group $A_6$