As title. the exponent of $G$ is the least number $n$ (if exists) such that $g^n=e$ holds for all $g\in G$ or $+\infty$.



Suppose that $G$ is a subgroup of $GL_n(\mathbb C)$ with finite exponent. Then is $G$ a finite group?
Since I was unable to find where else in MO this was answered, I'll answer this here. You can find a proof in chapter I, section 5 (pp. 83-84) of Algebra IV: Infinite Groups, Linear Groups by Kostrikin and Shafarevich; here's a Google books [link](https://books.google.com/books?id=O9cATtl-fowC&pg=PA83&lpg=PA83&dq=matrix+group+%22finite+exponent%22&source=bl&ots=OXyJkBami5&sig=t16MzHgZ_eGvkw6Zp9Om8uoZLgk&hl=en&sa=X&ei=qSoEVbHUEoLEggSWsIOQCA&ved=0CCwQ6AEwAg#v=onepage&q=matrix%20group%20%22finite%20exponent%22&f=false). 


