Problem 1 (yau-contest 2016)
Section: Algebra and Number Theory

Let $E$ be a linear space over $\mathbb{R}$, of finite dimension $n \ge 2$, equipped with a positive definite symmetric bilinear form $\langle\cdot,\cdot\rangle$. Let $u_1,u_2,\ldots,u_n$ be a basis of $E$. Let $v_1,v_2,\ldots,v_n$ be the dual basis, that is,

$$\langle u_i, v_j \rangle = \begin{cases} 1 & \text{if } i=j, \\ 0 & \text{if } i\ne j, \end{cases}$$
for all $i,j=1,2,\ldots,n$.

(a) Assume that $\langle u_i, u_j \rangle \le 0$ for all $1 \le i < j \le n$. Show that there is an orthogonal basis $u_1',u_2',\ldots,u_n'$ of $E$ such that $u_i'$ is a non-negative linear combination of $u_1,u_2,\ldots,u_i$, for all $i=1,2,\ldots,n$.

(b) With the same assumption as in Part (a), show that $\langle v_i, v_j \rangle \ge 0$ for all $1 \le i < j \le n$.

(c) Assume that $n \ge 3$. Show that the condition $\langle u_i, u_j \rangle \ge 0$ for all $1 \le i < j \le n$ does not imply that $\langle v_i, v_j \rangle \le 0$ for all $1 \le i < j \le n$.

Problem 2 (yau-contest 2016)
Section: Algebra and Number Theory

Let $d \ge 1$ and $n \ge 1$ be integers.

(a) Show that there are only finitely many subgroups $G \subseteq \mathbb{Z}^d$ of index $n$. Let $f_d(n)$ denote the number of such subgroups.

(b) Let $g_d(n)$ denote the number of subgroups $H \subseteq \mathbb{Z}^d$ of index $n$ such that the quotient group is cyclic. Show that $f_d(mn)=f_d(m)f_d(n)$ and $g_d(mn)=g_d(m)g_d(n)$ for coprime positive integers $m$ and $n$.

(c) Compute $g_d(p^r)$ for every prime power $p^r$, $r \ge 1$.

(d) Compute $f_d(20)$.

Problem 3 (yau-contest 2016)
Section: Algebra and Number Theory

Let $A$ be a complex $m\times m$ matrix. Assume that there exists an integer $N \ge 0$ such that $t_n=\operatorname{tr}(A^n)$ is an algebraic integer for all $n \ge N$. The goal of this problem is to show that the eigenvalues $a_1,\ldots,a_m$ of $A$ are algebraic integers.

(a) Show that there exist algebraic numbers $b_{ij} \in \mathbb{C}$, $1 \le i,j \le m$ such that

$$a_i^n = \sum_{j=1}^m b_{ij}\, t_{n+j-1}$$

for all $n \ge 0$ and all $1 \le i \le m$. In particular, $a_1,\ldots,a_m$ are algebraic numbers.

(b) Let $R$ be the ring of all algebraic integers in $\mathbb{C}$ and let $K$ be the field of all algebraic numbers in $\mathbb{C}$. Show that for $a \in K$, if $R[a]$ is contained in a finitely-generated $R$-submodule of $K$, then $a \in R$.

(c) Conclude that $a_1,\ldots,a_m$ are algebraic integers.

Problem 4 (yau-contest 2016)
Section: Algebra and Number Theory

Let $E$ be a Euclidean plane. For each line $l$ in $E$, write $s_l \in \operatorname{Iso}(E)$ for the reflection with respect to $l$, where $\operatorname{Iso}(E)$ denotes the group of distance-preserving bijections from $E$ to itself.

(a) Let $l_1$ and $l_2$ be two distinct lines in $E$. Find the necessary and sufficient condition that $s_{l_1}$ and $s_{l_2}$ generate a finite group.

(b) Let $l_1$, $l_2$ and $l_3$ be three pairwise distinct lines in $E$. Assume that $s_{l_1}$, $s_{l_2}$, and $s_{l_3}$ generate a finite group. Show that $l_1$, $l_2$, $l_3$ intersect at a point.

(c) Let $G$ be a finite subgroup of $\operatorname{Iso}(E)$ generated by reflections. Show that $G$ is generated by at most two reflections.

Problem 5 (yau-contest 2016)
Section: Algebra and Number Theory

Let $G$ be a finite group of order $2^n m$ where $n \ge 1$ and $m$ is an odd integer. Assume that $G$ has an element of order $2^n$. The goal of this problem is to show that $G$ has a normal subgroup of order $m$.

(a) Show that if $M$ is a normal subgroup of $G$ of order $m$, then $M$ is the only subgroup of $G$ of order $m$.

(b) Let $N$ be a normal subgroup of $G$ and let $P$ be a $2$-Sylow subgroup of $G$. Show that $P \cap N$ is a $2$-Sylow subgroup of $N$.

(c) Show that the homomorphism $G \to \{\pm 1\}$ carrying $g$ to $\operatorname{sgn}(\ell_g)$ is surjective. Here $\operatorname{sgn}(\ell_g)$ denotes the sign of the permutation $\ell_g : G \to G$ given by left multiplication by $g$.

(d) Deduce by induction on $n$ that $G$ has a normal subgroup of order $m$.