Does A8 contain an element of order 26? ( If X is a finite set, we can assume X

Question

Does A8 contain an element of order 26? ( If X is a finite set, we can assume X = {1, 2, . . . , n}. In this case we write Sn instead of SX. The following theorem says that Sn is a group. We call this group the symmetric group on n letters. Theorem 5.1 The symmetric group on n letters, Sn, is a group with n! elements, where the binary operation is the composition of maps.One of the most important subgroups of Sn is the set of all even permutations, An. The group An is called the alternating group on n letters.In general, the permutations of a set X form a group SX.)

Explanation / Answer

The possible orders of the elements of the alternating group A8 are {1, 2, 3, 4, 5, 6, 7, 15}.

A8 consists of only even permutations and hence the maximum order of any element of A8 can be 15 which is a small number so it's easy to check for each number between 1 and 15 that whether there exists an element of that order or not.

Example:

8 = 1 + 2 + 2+ 3, ignoring {1,3}, we get an even number of even numbers, so a permutation with this cycle structure is in A8.

It's order is lcm{1, 2, 2, 3} = 6.

Then the example would be (23)(45)(678) .

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