Taken off University of South Carolina Algebra PhD Qualifying Exam, January 2007
PROBLEM
Prove that a group with a proper subgroup of finite index has a proper normal subgroup of finite index.
PROOF
Let $G$ be a group and $H$ be a subgroup of finite index. Then $G/H=\{gH\}$ has finitely many elements. Let $G$ act on $G/H$ by $g(xH)=gxH$. This action clearly is well defined and gives rise to the homomorphism $\varphi:G\rightarrow S(G/H)$ where $S(G/H)$ is the permutation group of the elements of $G/H$. So $G/ker \varphi\simeq Im(G)\leq S(G/H)$, which is finite. Since $ker \varphi \triangleleft G$, we have that $G$ has a normal sungroup of finite index.
PROBLEM
Prove that a group with a proper subgroup of finite index has a proper normal subgroup of finite index.
PROOF
Let $G$ be a group and $H$ be a subgroup of finite index. Then $G/H=\{gH\}$ has finitely many elements. Let $G$ act on $G/H$ by $g(xH)=gxH$. This action clearly is well defined and gives rise to the homomorphism $\varphi:G\rightarrow S(G/H)$ where $S(G/H)$ is the permutation group of the elements of $G/H$. So $G/ker \varphi\simeq Im(G)\leq S(G/H)$, which is finite. Since $ker \varphi \triangleleft G$, we have that $G$ has a normal sungroup of finite index.
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