I was having some trouble understanding the following question:
Q: Let $G=\{1,-1\}$ endowed with the classical multiplication of integers.
Describe the left multiplication $\ell_{(1,-1)}$ by the element $(1,-1)$ in $G\times G$ as a permutation of the element of the set $G\times G$.
So I have been researching and I understand I have to write it as the product of disjoint cycles somehow but I have no idea how to apply this theorem to actual examples. If anyone could perhaps explain it in a simple way I would be very grateful.
In a group the function $l_g:x\mapsto gx$ is a permutation of the set of group elements.
The question asks you to describe this permutation in the particular case where the group is the direct product of the group $G$ described with itself and the particular $g$ in the above expression is the element (1,-1) in that group.
You can describe the permutation by showing the result of the permutation on each element of $G\times G$ (there are only four).
E.g. $l_{(1,-1)}:(1,1)\mapsto (1,-1)\ \ (1,-1)\mapsto (1,1)\ldots$ (complete the description by replacing the dots by the mappings for the remaining elements.)
Or as a cycle decomposition of the permutation:
$l_{(1,-1)}=((1,1),(1,-1))(\ldots)$ (again complete by replacing the dots with the content of the other cycle.)