Problem Set 1

Instructions: Write up your solutions using LaTeX and submit them on HuskyCT by September 11, 2022.

Problem 1: Let $\sigma=(13)(1235)(4567)$ and $\tau=(24)(35)(245)$.

1. Find $\sigma\tau$.
2. Find disjoint cycle decompositions of both $\sigma$ and $\tau$.
3. Write each of $\sigma$ and $\tau$ as products of transpositions.
4. Find the sign of $\sigma$ and $\tau$.

Problem 2: Define an ordering $\lesssim$ on the positive integers by saying that $n\lessim m$ if there is an injective homomorphism from $\Zn{n}\to\Zn{m}$.

1. Show that this relation is reflexive ($n\lessim n$ for all $n$), antisymmetric ($n\lessim m$ and $m\lesssim n$ implies $n=m$), and transitive ($n\lessim m$ and $m\lessim k$ implies $n\lessim k$.) These axioms mean that the positive integers are a partially ordered set under this order relation.

2. The meet $g$ of two elements $a$ and $b$ in a partially ordered set is an element that satisfies these two conditions:
   • $g\lessim a$ and $g\lessim b$.
   • If $h$ is any element satisfying $h\lessim a$ and $h\lessim b$, then $h\lessim g$.

   Prove that any two elements $m$ and $n$ have a meet.

3. Describe all of this fancy stuff in a simpler way.

4. (Extra) What happens if, instead of considering injective homomorphisms from $\Zn{n}$ to $\Zn{m}$, we consider surjective ones?

Problem 3: If $H$ is a subgroup of $G$, then the normalizer $N_{G}(H)={g\in G: gHg^{-1}=H}$ and the centralizer $C_{G}(H)={g\in G : gh=hg\quad \forall h\in H}$.

1. Let $G=\GL_{2}(\R)$ and let $H=\mathrm{Aff}(\R)$ be the affine group consisting of two by two matrices of the form $\left(\begin{matrix} x & y\cr 0 & 1\end{matrix}\right)$ where $x\not=0$. Show that the centralizer of $H$ in $G$ consists of matrices $aI$ where $a\not=0$ and $I$ is the identity matrix, and the normalizer consists of the upper triangular matrices with nonzero diagonal entries. What about the same question for $\GL_{2}(\Q)$? Does the field matter for this?
2. Let $G=\GL_{2}(\R)$ and $H$ be the subgroup of diagonal matrices. Show that $H$ is its own centralizer, and its normalizer consists of diagonal matrices and “anti-diagonal” matrices $\left(\begin{matrix} 0 & a\cr b & 0\end{matrix}\right)$ in $G$.
3. View $S_{n}$ as $\mathrm{Sym}(\Zn{n})$. Given integers $a$ and $b$, with $\gcd(a,n)=1$, let $\sigma_{a,b}:\Zn{n}\to\Zn{n}$ be the map $\sigma_{a,b}(x)=ax+b\pmod{n}$. Show that $\sigma_{a,b}$ is in the normalizer of the cyclic subgroup $H$ generated by $(123\ldots n)$ and, conversely, any element of the normalizer of this subroup is $\sigma_{a,b}$ for some $a$ and $b$. Conclude that the normalizer in $S_{n}$ of $H$ is the affine group of $\Zn{n}$.

Problem 4: Let $M$ be the group with two generators $u$ and $v$ satisfying $u^2=v^8=1$ and $vu=uv^{5}$. This group has order $16$ (can you verify this?). $M$ has three subgroups of order $8$: $\langle u,v^2\rangle$, $\langle v\rangle$, and $\langle uv \rangle$. Every proper subgroup is contained in one of these three groups.

1. Draw the lattice of subgroups of $M$.
2. Prove that the group generated by $v^{4}$ is normal in $M$.
3. Find the lattice of subgroups of $M/\langle v^{4}\rangle$ inside that of $M$ using the lattice isomorphism theorem.

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