Understanding an elementary proof from Dummit and Foote’s Abstract Algebra

Question

I am reading the proof of Proposition 5 in Section 3.1 of Dummit and Foote's Abstract Algebra. Can someone please help me understand why $ng^{-1}\cdot 1\in ng^{-1}N$ implies $ng^{-1}\in g^{-1}N$?

Answer 1

In the previous line you showed $g^{-1} N = n g^{-1} N$.