cmi-entrance 2012 QB7

cmi-entrance · India · pgmath 10 marks Groups Subgroup and Normal Subgroup Properties

(i) Let $G = G L \left( 2 , \mathbb { F } _ { p } \right)$. Prove that there is a Sylow $p$-subgroup $H$ of $G$ whose normalizer $N _ { G } ( H )$ is the group of all upper triangular matrices in $G$.
(ii) Hence prove that the number of Sylow subgroups of $G$ is $1 + p$.

(i) Let $G = G L \left( 2 , \mathbb { F } _ { p } \right)$. Prove that there is a Sylow $p$-subgroup $H$ of $G$ whose normalizer $N _ { G } ( H )$ is the group of all upper triangular matrices in $G$.\\
(ii) Hence prove that the number of Sylow subgroups of $G$ is $1 + p$.

Paper Questions

QA1 QA10 QA11 QA12 QA13 QA14 QA15 QA2 QA3 QA4 QA5 QA6 QA7 QA8 QA9 QB1 QB10 QB2 QB3 QB4 QB5 QB6 QB7 QB8 QB9