cmi-entrance 2021 Q11

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

Let $G$ be a finite group and $X$ the set of all abelian subgroups $H$ of $G$ such that $H$ is a maximal subgroup of $G$ (under inclusion) and is not normal in $G$. Let $M$ and $N$ be distinct elements of $X$. Show the following:
(A) The subgroup of $G$ generated by $M$ and $N$ is contained in the centralizer of $M \cap N$ in $G$.
(B) $M \cap N$ is the centre of $G$.

Let $G$ be a finite group and $X$ the set of all abelian subgroups $H$ of $G$ such that $H$ is a maximal subgroup of $G$ (under inclusion) and is not normal in $G$. Let $M$ and $N$ be distinct elements of $X$. Show the following:\\
(A) The subgroup of $G$ generated by $M$ and $N$ is contained in the centralizer of $M \cap N$ in $G$.\\
(B) $M \cap N$ is the centre of $G$.

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