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

Statement
Suppose $$G$$ is a fact about::finite simple non-abelian group and $$A$$ is a proper abelian subgroup of $$G$$. Then, $$|G| > |A|^2$$.

Facts used

 * 1) uses::Finite simple non-abelian group has order greater than product of order of proper subgroup and its centralizer

Proof
The proof follows from fact (1), and the fact that if $$A$$ is an abelian subgroup, $$|C_G(A)| \ge |A|$$.

Journal references

 * Theorem 2.1 (Page 909) of