Math 120A HW 8
Due Tuesday, December 9.
- Let $\phi : G \to G'$ be a homomorphism of groups.
-
Show that if $h \in \ker(\phi)$ and $g \in G$, then $ghg^{-1} \in
\ker(\phi)$. Prove it directly from the definitions of "kernel" and "homomorphism" instead of relying on theorems saying that the kernel is a normal subgroup and that normal subgroups are closed under conjugation.
- Show that if $G'$ is abelian and $a,b \in G$ then $aba^{-1}b^{-1}
\in \ker(\phi)$. (We call $aba^{-1}b^{-1}$ the commutator of $a$
and $b$. It equals the identity if and only if $a$ and $b$ commute with
each other.)
- Recall that $SL_2(\mathbb{R})$ is the group of $2 \times 2$ real
matrices of determinant $1$, and recall that it is a normal subgroup of
$GL_2(\mathbb{R})$.
-
Defining the matrices
$$ g = \begin{pmatrix}
-1 & 0\\
0 & 1
\end{pmatrix} \quad\text{and} \quad
h = \begin{pmatrix}
0 & 1\\
-1 & 0
\end{pmatrix},$$
show that $g \in GL_2(\mathbb{R})$ and $h \in SL_2(\mathbb{R})$ but $gh \ne hg$.
- Find another matrix $h' \in SL_2(\mathbb{R})$ such that $gh = h'g$.
- Let $H$ be the cyclic subgroup of $\mathbb{Z} \times \mathbb{Z}$ generated by $(1,1)$.
- Give a condition on the numbers $i,j,k,l \in \mathbb{Z}$ that is necessary and sufficient for the two cosets $(i,j) + H$ and $(k,l) + H$ to be equal. Your condition should not mention $H$.
- Show that the coset $(0,1) + H$ is a generator of the factor group $(\mathbb{Z} \times \mathbb{Z}) / H$.
- Show that the coset $(0,1) + H$ has infinite order in the factor group $(\mathbb{Z} \times \mathbb{Z}) / H$. (It is not enough to show that the element $(0,1)$ has infinite order in $\mathbb{Z} \times \mathbb{Z}$.)
Hint: for parts b and c, you can use the general fact that $n(a
+H) = (na) + H$ when $a$ is an element of an abelian group, $H$ is a
subgroup, and $n$ is an integer. When $n = 2$ this fact follows from the definition of $(a+H) +
(a+H)$, and the proof of the general case uses induction.
- Let $G$ be a group and let $H$ be a normal subgroup of $G$.
For cosets $aH, bH \in G/H$ recall that we defined their product by $(aH)(bH) = (ab)H$.
Another natural definition would be $(aH)(bH) = \{cd : c \in aH \mathbin{\And} d \in bH\}$.
Show that these definitions are equivalent; in other words, show that
$$ (ab)H = \{cd : c \in aH \mathbin{\And} d \in bH\}.$$
Where did you use normality in your proof?
- Let $G$ be a group. For $x,y \in G$ we say that $y$ is conjugate to $x$ if $y = i_g(x)$ for some $g \in G$, where the inner automorphism $i_g$ of $G$ is defined by $i_g(x) = gxg^{-1}$.
- Prove that conjugacy is an equivalence relation (e.g. for symmetry, prove that if $y$ is conjugate to $x$ then $x$ is conjugate to $y$.)
- The equivalence classes of this equivalence relation are called conjugacy classes of $G$. List the conjugacy classes of $S_3$. (I do not require you to prove that your list is correct.)
- Let $H$ be the subgroup of $GL_2(\mathbb{R})$ consisting of all diagonal matrices. Prove that $H$ is not a normal subgroup of $G$. Hint: a conjugate of a diagonal matrix is diagonalizable, but not necessarily diagonal.
Optional practice problems from the book (these will not be graded): Section 13 Exercises 33, 37, 49; Section 14 Exercises 5, 9, 23, 27, 31, 38.