Finite simple non-abelian group has order greater than product of order of proper subgroup and its centralizer

Statement
Suppose $$G$$ is a finite simple non-abelian group and $$A$$ is a proper subgroup of $$G$$. Then, $$|G| > |A||C_G(A)|$$.

Applications

 * Every proper abelian subgroup of a finite simple non-abelian group has order less than its squareroot

Journal references

 * Theorem 2.1 (Page 909) of