What are interesting / non-trivial examples of smooth connected closed manifolds that are direct products or involve direct products? I am especially interested in orientable manifolds. Say, an $n$-torus $T^n$ is a direct product of $n$ copies of a circumference $S^1$. One can build a 3-manifold from a surface of genus $g$ as $M=M^2_g\times S^1$, […]

I don’t know about the Sylow Theorems. But I have been wondering about a proof of the fact that a group or order $pq$ where $p$ and $q$ are distinct primes must be cyclic. I can’t quite work out the details, but here is the general idea. I would like help with filling in details. […]

Intereting Posts

Is the set theory (ZF) a structure?
$G$ be a non-measurable subgroup of $(\mathbb R,+)$ ; $I$ be a bounded interval , then $m^*(G \cap I)=m^*(I)$?
Projective but not free (exercise from Adkins – Weintraub)
Maximum of $a_1 \cdot a_2 \cdots a_n$ given $a_1 + \cdots + a_n = 1000$?
Elementary number theory – prerequisites
How to cut a cube into an icosahedron?
Fast Fourier Transformation, explanation of $A(x)=A_0(x^2)+x A_1(x^2)$
Is there a simpler approach to these system of equations?
Vector space bases without axiom of choice
Computing $\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right) \, dx$
Find all positive values for j,k,l such that j, k, l are positive integers and (j-k)|l, (k-l)|j, (l-j)|k.
Inverse of Symmetric Matrix Plus Diagonal Matrix if Square Matrix's Inverse Is Known
Examples of patterns that eventually fail
Understanding dependency graph for a set of events
Is indefinite integration non-linear?