cmi-entrance 2022 Q14

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

Let $G$ be a finite group that has a non-trivial subgroup $N$ (i.e. $\left\{ 1 _ { G } \right\} \neq N \neq G$ ) that is contained in every non-trivial subgroup of $G$. Show that
(A) $G$ is a $p$-group for some prime number $p$;
(B) $N$ is a normal subgroup of $G$.

Let $G$ be a finite group that has a non-trivial subgroup $N$ (i.e. $\left\{ 1 _ { G } \right\} \neq N \neq G$ ) that is contained in every non-trivial subgroup of $G$. Show that\\
(A) $G$ is a $p$-group for some prime number $p$;\\
(B) $N$ is a normal subgroup of $G$.

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