The question is does $A_8$ have an element of order 26? My knowledge on alternating groups is very limited. I understand that the order of $A_8$ is $\frac{8!}{2}$ but that is essentially all. My one thought is maybe we use the theorem on cyclic subgroups, where the orders relate to $gcd$. Even that I am not sure how I would use though.
The teacher has not explained this and we are using the free book Abstract Algebra by Judson, which does not discuss orders of elements in alternating subgroups, only the order of the alternating subgroup.
I am sorry if I have not provided enough information, I feel like with this question I am not sure where to begin.
Edit: We have not done Lagrange's yet
Best Answer
The order of an element must divide the order of the group.
26 does not divide 8!/2, because 13 divides 26 but not 8!.
Therefore $A_8$ cannot contain an element of order 26.