Math 120A HW 9

Due Wednesday, March 11.
  1. For this problem it will be helpful to use the fundamental theorem of finite abelian groups.
    1. List all abelian groups of order $252$ (${}= 2^2 3^2 7$) up to isomorphism.
    2. Prove that every abelian group of order $252$ has an element of order $42$.
  2. Is the direct product $A_4 \times \mathbb{Z}_2$ isomorphic to $S_4$? Justify your answer.
  3. Let $G$ and $G'$ be groups and let $\phi$ be a homomorphism from $G$ to $G'$. Prove that $\operatorname{ker}(\phi)$ is a subgroup of $G$, without using the more general fact that the inverse image of a subgroup under a homomorphism is a subgroup.
  4. Let $p$ be a prime number, let $G$ be a group, and let $\phi : \mathbb{Z}_p \to G$ be a homomorphism. Prove that $\phi$ is either injective or trivial. (Recall that a homomorphism is called trivial if its range is trivial.)
  5. Consider the following two subgroups of $\operatorname{GL}_2(\mathbb{R})$. (You may assume that they are subgroups.) $$\begin{align} H_1 &= \left\{ \begin{pmatrix} x & 0 \\ 0 & x \end{pmatrix} : x \in \mathbb{R}^* \right\}\\ H_2 &= \left\{ \begin{pmatrix} x & 0 \\ 0 & y \end{pmatrix} : x,y \in \mathbb{R}^* \right\}. \end{align}$$
    1. Is $H_1$ normal in $\operatorname{GL}_2(\mathbb{R})$? Justify your answer.
    2. Is $H_2$ normal in $\operatorname{GL}_2(\mathbb{R})$? Justify your answer.
    (It may help to recall that for a subgroup $H$ of a group $G$, an equivalent condition for normality is "$aHa^{-1} \subseteq H$ for all $a \in G$," or in other words, "$aha^{-1} \in H$ for all $a \in G$ and $h \in H$.")
  6. Let $G$ be a group and let $H$ be a subgroup of $G$.
    1. Let $a \in G$. Prove that $aHa^{-1}$ is a subgroup of $G$, where $aHa^{-1} = \{aha^{-1} : h \in H\}$.
    2. Prove that if $G$ has no other subgroups with the same order as $H$, then $H$ is normal in $G$.