4.77 (1973)

In 1972, A. Rudvalis discovered a new simple group $R$ of order $2^{14} \cdot 3^3 \cdot 5^3 \cdot 7 \cdot 13 \cdot 29$. He has shown that $R$ possesses an involution $i$ such that $C_R(i) = V \times F$, where $V$ is a 4-group (an elementary abelian group of order 4) and $F \cong \text{Sz}(8)$.
$\qquad$ a) Show that $R$ is the only finite simple group $G$ that possesses an involution $i$ such that $C_G(i) = V \times F$, where $V$ is a 4-group and $F \cong \text{Sz}(8)$.
$\qquad$ b) Let $G$ be a non-abelian finite simple group that possesses an involution $i$ such that $C_G(i) = V \times F$, where $V$ is an elementary abelian 2-group of order $2^n, n \geqslant 1$, and $F \cong \text{Sz}(2^m), m \geqslant 3$. Show that $n = 2$ and $m = 3$.