Let $G$ be a finite group of odd order and $1 < d < | G |$ be a divisor of $| G |$. Assume that $G$ has exactly three subgroups $H _ { 1 } , H _ { 2 }$ and $H _ { 3 }$ of order $d$. Suppose that $H _ { 1 }$ is not normal in $G$. For each $i = 1,2,3$, let $N _ { i }$ denote the normalizer of $H _ { i }$. Let $S : = \left\{ H _ { 1 } , H _ { 2 } , H _ { 3 } \right\}$. (A) (4 marks) For each $g \in G$ let $s _ { g }$ denote the cardinality of the set $\left\{ H \in S \mid g H g ^ { - 1 } = H \right\}$. Show that $\sum _ { g \in G } s _ { g } = \left| N _ { 1 } \right| + \left| N _ { 2 } \right| + \left| N _ { 3 } \right|$. (B) (6 marks) Show that $G \neq N _ { 1 } \cup N _ { 2 } \cup N _ { 3 }$.
Let $G$ be a finite group of odd order and $1 < d < | G |$ be a divisor of $| G |$. Assume that $G$ has exactly three subgroups $H _ { 1 } , H _ { 2 }$ and $H _ { 3 }$ of order $d$. Suppose that $H _ { 1 }$ is not normal in $G$. For each $i = 1,2,3$, let $N _ { i }$ denote the normalizer of $H _ { i }$. Let $S : = \left\{ H _ { 1 } , H _ { 2 } , H _ { 3 } \right\}$.\\
(A) (4 marks) For each $g \in G$ let $s _ { g }$ denote the cardinality of the set $\left\{ H \in S \mid g H g ^ { - 1 } = H \right\}$. Show that $\sum _ { g \in G } s _ { g } = \left| N _ { 1 } \right| + \left| N _ { 2 } \right| + \left| N _ { 3 } \right|$.\\
(B) (6 marks) Show that $G \neq N _ { 1 } \cup N _ { 2 } \cup N _ { 3 }$.