# What exactly does ${\rm PGL}(n,K)$ represent?

Suppose $V:=GL(n,K)$ denotes the general linear group and $V_I$ denotes center of $V$. Then what exactly does $V/V_I$ represent? I would be appreciate if you give a similar good example for better understanding. (The origin of this question is General projective transformations ${\rm PGL}(n,K)$ and special projective transformations ${\rm PSL}(n,K)$)

• The identity element of a vector space is the zero vector. It spans a trivial subspace. In other words $V_I = \{0\}$ and $V / V_I$ is generated by the equivalence relation on $V$ where $u\sim v \iff u-v\in V_I$ i.e $u=v$. So for all intents and purposes $V = V / V_I$. – Alvin Lepik Nov 22 '17 at 8:11
• So why $PGL(n,K)=GL/ z$ where $z$ is the center of $GL(n,K)$. isn't center of $GL(n,K)$ equal to $V_I$? – C.F.G Nov 22 '17 at 9:02
• The center of a group is the subset of those elements that commute with all elements in the group. What makes you think a center must always be trivial? – Alvin Lepik Nov 22 '17 at 9:14
• $PGL(n,K) = GL/z$ is merely the definition of what a projective linear group is (according to wiki). I don't understand what your question(s) are. – Alvin Lepik Nov 22 '17 at 9:17
• According to this post math.stackexchange.com/questions/960342/… $V_I=a I$ – C.F.G Nov 22 '17 at 9:36