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?