0
$\begingroup$

In my study of group homomorphism I find a question like this.

Let $Q\colon \mathbb Z\to S_8$ be group homomorphism such that $Q(1)=(1,4,2,6)(2,5,7)$. Then find $\ker(Q)$ and $Q(21)$.

In my mind I know that $\ker(Q)$ are those elements of $\mathbb Z$ which map to the identity of $S_8$ .How I can work with this concept to get those elements? Much more difficult I get is to find $Q(21)$.Thank again

  • 3
    First find the smallest positive integer $n$ such that $Q(n)$ is the identity permutation. Remember that the group operation is addition of integers on one side and composition of permutations on the other. $Q(2)=Q(1+1)$,$Q(3)=Q(2+1),\ldots$2011-12-25

2 Answers 2