# Tag Info

Accepted

### Does every sequence of group epimorphisms (between finitely generated groups) contain a stable subsequence?

Yes. This is just (metrizable) compactness in the space of normal subgroups of $G$. It is enough to assume that $G$ is countable (finitely generated plays no role). Namely, let $N(G)\subset 2^G$ be ...
• 57.9k

### Are large powers of polynomials linearly independent?

We have used this problem for our Student Olympiad in Algebra at Moscow State University (in Russian, Пятнадцатая олимпиада, задача 8). So, here is a completely elementary solution. Exercise 1. Show ...
• 3,892
Accepted

### Analogous results in geometric group theory and Riemannian geometry?

I think Cheeger's inequality is a good example. Riemannian geometry version Let $M$ be a closed Riemannian $n$-manifold. Say that a $n-1$ dimensional submanifold $N$ separates $M$ if the complement of ...
• 28.4k
Accepted

### When are two semidirect products of two cyclic groups isomorphic

The paper of Basmaji, "On the isomorphisms of two metacyclic groups" (Proc AMS 1969) gives a complete answer to the question of when two finite metacyclic groups with the same $m$ and $n$ ...
• 8,534

### Analogous results in geometric group theory and Riemannian geometry?

Here is a very classical example. As stated in the comments, Gromov was an early proponent of importing ideas from geometry to group theory, but already thirty years earlier there was work in this ...
Accepted

### Morita equivalences and centers of some algebras

The answer is that in your matrix $\left(\begin{smallmatrix} 0&x_0\\0&0\end{smallmatrix}\right)$, the $x_0$ denotes the isomorphism of modules given by left multiplication by $x_0$, so it ...
• 8,534
Accepted

### Where has this structure been observed?

This is an infinite commutative diagram on $M$ (viewed as a category with a single object $\bullet$). $\require{AMScd}$ \begin{CD} \vdots @. \vdots @. \vdots\\ @VVR_y(0,2)V @VVR_y(1,2)V @VVR_y(2,2)V\...
• 104k
Accepted

### Question to limit groups (over free groups)

You need to prove the following folklore lemma, which is well known to researchers in the field but perhaps not written down anywhere. The proof is a nice exercise. Folklore lemma: Let $S$ be a ...
• 23.5k

### Concept associated to the Eudoxus reals

This method can be used to construct the fields $\mathbb{Q}_p$ and the ring $\mathbb{A}_{\mathbb{Q}}$ of adeles over $\mathbb{Q}$. See T.D.J. Hermans' Bachelor's thesis: https://www....
• 3,786
Accepted

Here is an example that shows that you can expect things to get as bad as it goes (I learned about this algebra from the wonderful article The Non-Commutative Gröbner Freaks by Green, Mora, and ...
• 15.9k
Accepted

• 5,732

### Are large powers of polynomials linearly independent?

$\require{AMScd} \require{enclose}$EDIT : As noted by Zach Teitler, the argument below only proves that for $m\gg0$, the family $\left\{P_1^{\otimes m}, \dotsc, P_k^{\otimes m} \right\}$ is a free ...
• 6,960

### An algebra map between Hopf algebras that does not commute with the counit

Let $H$ be the Hopf algebra of functions on an algebraic group $G$. The map $\phi$ defines a map of algebraic varieties $\hat{\phi}:G\rightarrow G$. The counit condition you try to impose is ...
• 11.8k