my aim : Why $GL(V)/Z(GL(V))$ is termed as projective general linear group?
The reason seen in books says This group acts on the projective set/space faithfully.
But, there can be many groups acting faithfully on projective space, can we call them also as projective general linear group?
I confused, and not understood the reasonings. I faced many questions from it:
Let $V$ be $n$-dimensional vector space over $F$ and $V^*$ denote the collection of one dimensional subspaces of $V$, it is called projective space.
For $V^*$, we can first consider its symmetric group $Symm(V^*)$- the set of all the bijections from $V^*$ to itself. If we put some structure on $V^*$, and if we look bijections preserving the structure on $V^*$, then we obtain a subgroup of $Symm(V^*)$
Question 1. What stuctural properties on $V^*$ the group $GL(V)/Z(GL(V))$ preserves?
question 2. Can we say that, $GL(V)/Z(GL(V))$ is precisely the set of all bijections from $V^*$ to itself which preserve certain structure on $V^*$?
Question 3. Is there some notion like Projective group associated to $V^*$ instead of Projective Linear Group? If yes, how that group differs from $GL(V)/Z(GL(V))$?
Question 4. Since passing from $V$ to $V^*$, we have lost linearity: $V^*$ is not a vector (linear) space. Then what the term linear refers to in Projective Linear Group?
[Please point out, if questions are not clear.]