Proposition 2.4 in Jonathan S. Golan "Linear Algebra a beginning graduate student ought to know"

Question

The proposition and its proof is given below:

My question is: why the set $B$ becomes ${a a_{1}, ... ,a a_{q-1} }$? the idea of the proof is not very clear for me regarding to this step. should not we change the name of $B$?

Answer 1

Multiplication by a nonzero element in a field is an injective map from $B\to B$, and since $B$ is finite, it is a permutation of $B$. It simply rearranges the elements of $B$.

Answer 2

The $a_i$s are distinct non-zero members of the field. Each non-zero element of the field is invertible so if $aa_i=aa_j$ then $a_i=a_j$ which would contradict the uniqueness of the $a_i$s. Therefore $aB=\{aa_1, aa_2, ..., aa_{q-1}\}$.

A field is also closed under multiplication so the unique elements of $aB$ have to be the same $q-1$ non-zero elements of $B$. Therefore, $aB=B$ so we don't have to rename the set.