Questions askedCategory: 6111Presenter for Sep 1st
Yu Staff asked 2 years ago

Here is the list of extra problems for the Sep 1st recitation.

Please comment below your name (initial, or only first name is fine) and the number of the problem you want to present in Thursday’s recitation.

e.g. # 3, Yu

  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 $2$.
    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 $p$.

Kirollos Masood replied 2 years ago

Kirollos: 4