To which group H is the Quotient Group $\frac{GL_n(Z_p) }{SL_n(Z_p)}$ isomorphic? So I need to find a surjective homomorphism  $\phi:  GL_n(\Bbb Z_p) \to H $ s.t. $ker\phi = SL_n(\Bbb Z_p)$ and use the First Isomorphism Theorem to find $\phi' :\frac{GL_n(Z_p) }{SL_n(Z_p)} \to H $ I'm not sure how to find $\phi$

Define a mapping f:Gln(Zp) - U(n)(i.e Z(p)/{0}) such that f(A) = det(A) for all A in Gln(Zp) then it is an onto homomorphism with kernel = SLn(Zp).(Kernel={A in Gln(zp) : f(A)(det(A)) = identity of U(n) (which is 1)}= Sln(Zp) . Now apply first isomorphism theorem - the quptient group GLn(Zp)/SLn(Zp) is isomorphic to U(p).
Order(Gln(Zp)/Sln(Zp))=O(U(n))=p-1 if p is prime .
