Hello everyone. I am asked to find the size of the group and would like to verify my work.
This group acts on the vector space with basis . From this, we have the orbit-stabilizer theorem: if , then , that is, the size of is the size of the orbit of times the size of the stabilizer of . Take . If , can possibly be any nonzero vector. There are 48 of these. Therefore, . Then, if , we must have equal to another vector not a multiple of , since is invertible. In other words, cannot be . There are 42 of these, so .
We conclude that .
Comments or suggestions?