midterm样题

1. (Show details of your computation). Let $M=\mathbb{Z}\cdot e_1\oplus \mathbb{Z}\cdot e_2$ be a free abelian group of rank 2 . Let $M^{\prime }=⟨2e_1+20e_2,2e_1+30e_2⟩$ be a subgroup. Find a new basis $\left(f_1,f_2\right)$ of $M$ so that $M^{\prime }$ can be written as $⟨a_1{f}_1,a_2{f}_2⟩$ with $a_1\mid a_2$.

2. Let $G,H$ be two finite cyclic groups. Prove that $G\times H$ is a cyclic group if and only if $(|G|,|H|)=1$.

\begin{proof} Assume that $⟨g⟩=G$, $⟨h⟩=H$.

If $(|g|,|h|)=1$, then $(g,h)\in G\times H$ generates $G\times H$ and so it is cyclic.

If $G\times H$ is cyclic, then there exists $(k_1,k_2)$ generates $G\times H$. The order of

$$(k_1,k_2)=\mathrm{lcm}(|k_1|,|k_2|)=|k_1||k_2|/(|k_1|,|k_2|)=|k_1||k_2|$$

and so $(|G|,|H|)=1$. \end{proof}

3. Let $G$ be a finite group with center $C(G)$. Show that $G/C(G)$ is abelian if and only if for each $x\notin C(G)$, the conjugacy class of $x$ is contained in the coset $C(G)x$.

\begin{proof} If $G/C(G)$ is abelian, then for any $x\notin C(G)$ we have ${\overline{x}}^{\overline{y}}=\overline{x}$ where $\overline{\cdot }:G\to G/C(G)$ is the quotient map and so $x^y\in C(G)x$.

If for each $x\notin C(G)$, the conjugacy class of $x$ is contained in the coset $C(G)x$, then for any $y\in G$, $x^y=cx$ for some $c\in C(G)$. It deduces that \overline{x^y}=\overline x^\overline y=\overline x and so $G/C(G)$ is abelian. \end{proof}

4. Let $G$ be a group with two subgroups $H_1,H_2$ such that $H_1⊈H_2,H_2⊈H_1$. Show that the union $H_1\cup H_2$ is never a subgroup of $G$.

\begin{proof} Since $H_1⊈H_2,H_2⊈H_1$, then there exists $h_1\in H_1\setminus H_2$ and $h_2\in H_2\setminus H_1$. If $H_1\cup H_2$ is a subgroup, then $h_1{h}_2$ is contained in either $H_1$ or $H_2$, neither is impossible. \end{proof}

5. Show that there does not exist group isomorphisms between $\mathbb{Q}$ and $\mathbb{Q}/\mathbb{Z}$.

\begin{proof} Assume that $\phi :\mathbb{Q}\to \mathbb{Q}/\mathbb{Z}$ is a group isomorphism, then $\phi (x_1+x_2)=\phi (x_1)+\phi (x_2)$ for any $x_1,x_2\in \mathbb{Q}$.

Assume that $\phi (1)=\frac{p}{q}+\mathbb{Z}$, then $\phi (q)=0+\mathbb{Z}$ and so $\phi (1/2+q)=\phi (1/2)+\phi (q)=\phi (1/2)$, $\phi$ is not injective and so it is not an isomorphism. \end{proof}

6. In the following problem, you need to explain why your answer is right.

• (a) Construct a commutative ring which has only one prime ideal.
• (b) Construct a commutative ring with finite elements which has precisely two prime ideals.
• (c) (maybe difficult) Construct a commutative ring with infinite elements which has precisely two prime ideals.

\begin{proof} a) $R={\mathbb{Z}}_p$ has only one prime ideals are $\{0\}$.

b) ${\mathbb{Z}}_{p^2}$.

c) $k[[x]]$.

7. (a) Suppose G is a group of order 42. Show it has a normal Sylow subgroup.

(b) Suppose G is a group of order 30. Show it has a normal Sylow subgroup.

\begin{proof} a) $n_7=1$

b) $n_5\in \{1,6\}$ and $n_3\in \{1,10\}$. If $n_5=6$ and $n_3=10$, then $|G|⩾5\times 5+2\times 10>30$, contradiction. \end{proof}

8. Suppose $G$ is a group of order 6 .

• (a) Show it has a normal Sylow subgroup.
• (b) Show there are only two possible groups (up to isomorphism) of order 6. You need to prove it.

\begin{proof} a) By Sylow theorems, $n_3=1$ and so $H⊲G$ with $H\simeq C_3$.

b) Since $|G|=6$, by Cauchy’s theorem there exist $g,h\in G$ such that $|g|=3$ and $|h|=2$. Furthermore, $⟨g⟩=H$ and $g^h\in \{g,g^2\}$. Let $H=⟨h⟩$ and $K=⟨g⟩$. Then $H\cap K=\{1\}$.

If $g^h=g$, then $H⊲G$ and $G=H\times K\simeq C_2\times C_3\simeq C_6$.

If $g^h=g^2$, then $G=⟨g⟩:⟨h⟩$. Define $\phi :g\mapsto (123),h\mapsto (12)$, then $G\simeq S_3$. \end{proof}

9. Let $G$ be a group of order $pqr$ with $p,q,r$ three distinct prime numbers. Prove it has a normal Sylow subgroup.

Easy.