Question

Let G be a group of order 112:132 . How many 11-sylow subgroups and 13 sylow subgroups are there in G?

Answer 1

Let $G$  be a group of order $11^2:13^2 .$  

It follows from Third Sylow Theorem that for the number $s_{11}$ of sylow $11$-subgroups we have that $s_{11}\equiv 1\mod 11$ and $s_{11}$ divides $13^2.$ Therefore, $s_{11}\in\{1,13,169\}\cap\{1,12,23,34,45,56,67,78,89,100,111,122,133,144,155,166,...\}\\=\{1\}.$

We conclude that $s_{11}=1,$ and hence there is a unique sylow $11$-subgroup in $G$.

It follows from Third Sylow Theorem that for the number $s_{13}$ of sylow $13$-subgroups we have that $s_{13}\equiv 1\mod 13$ and $s_{13}$ divides $11^2.$ Therefore, $s_{13}\in\{1,11,121\}\cap\{1,14,27,40,53,66,79,92,105,118,131,...\}=\{1\}.$

We conclude that $s_{13}=1,$ and hence there is a unique sylow $13$-subgroup in $G$.