Recent questions tagged cmi2024-math-part-b

Let $n \geq 3$ be an integer. Write $D_{2 n}$ for the dihedral group with $2 n$ elements. Show that the automorphism group of $D_{2 n}$ has at most $n \varphi(n)$ element...

Let $\mathrm{SL}(2, \mathbb{R})$ be the group of $2 \times 2$ matrices with real entries and determinant 1 , endowed with the subspace topology of $\mathbb{R}^{4}$. Consi...

Let $G$ be a finite group of odd order and $1<d<|G|$ be a divisor of $|G|$. Assume that $G$ has exactly three subgroups $H_1, H_2$ and $H_3$ of order $d$. Suppose that $H...

Let $X$ be a non-empty finite set and let $R$ be the ring of $\mathbb{Z}$-valued functions on $X$, with pointwise addition and multiplication. Let $S$ be an additive subg...

Let $A(X), B(X)$ be non-zero polynomials in $\mathbb{C}[X]$ such that $0 \leq \operatorname{deg} A \leq \operatorname{deg} B-2$ and $A(X)$ and $B(X)$ do not share any roo...

Let $f: \mathbb{R} \longrightarrow \mathbb{R}$ be a continuous function such that $f(r)=f\left(r+\frac{1}{n}\right)$ for each $r \in \mathbb{Q}$ and each positive integer...

Let $p$ be a prime number. Let $n \geq 2$ be an integer. Let $V$ be an $n$-dimensional $\mathbb{F}_{p}$-vector space. Determine, with a proof, the number of two-dimension...

Let $R$ be the ring of all the real-valued functions on $\mathbb{N} \times \mathbb{N}$. Show that $R$ contains a subring isomorphic to the polynomial ring $\mathbb{R}[X, ...

Let $M_{n}(\mathbb{R})$ be the space of $n \times n$ matrices with real entries, identified with the euclidean space $\mathbb{R}^{n^{2}}$. Let $X$ be a compact subset of ...

Fix $0<\lambda<1$. Choose $\epsilon>0$ such that $\epsilon+\frac{\lambda^{2}}{4} \leq \frac{\lambda}{2}$. Consider the metric space\[X:=\left\{\psi \in \mathcal{C}^{1}([-...