Extra problems 1

Here is a list of extra problems for Thursday September 1st recitation class.

  1. Let $G$ be a group and let $H$ be a subgroup of finite index.
    Then show that there exists a normal subgroup $N$ of $G$ such that $N$ is of finite index in $G$ and $N\subset H$.
  2. If $G$ is a group and $H$ and $K$ are two subgroups of finite index $m$ and $n$, respectively, in $G$,
    then prove that $\lcm(m,n) \leq |G: H \cap K| \leq mn$.
  3. Let $G$ be a finite group of order $n$ and suppose that $p$ is the smallest prime number dividing $n$.
    Then show that any subgroup of index $p$ is a normal subgroup of $G$.
  4. Let $H$ be a subgroup of order $2$ in $G$.
    Let $N_G(H)$ be the normalizer of $H$ in $G$ and $C_G(H)$ be the centralizer of $H$ in $G$.

    1. Show that $N_G(H)=C_G(H)$.
    2. If $H$ is a normal subgroup of $G$, then show that $H$ is a subgroup of the center $Z(G)$ of $G$.
  5. Let $G$ be a group. Suppose that the order of nonidentity element of $G$ is $2$.
    Then show that $G$ is an abelian group.
    1. Prove that $\Q$ has no proper subgroups of finite index.
    2. Prove that $\Q/\Z$ has no proper subgroups of finite index.
  6. Let $p$ be a prime number. Show that any group of order $p^2$ contains a normal subgroup of order $p$.

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