# Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

2,311 questions
Filter by
Sorted by
Tagged with
50 views

### Centralizer bound for irreducible representations of $\operatorname{SU}_n(\mathbb{C})$

Let $G$ be a finite group, $χ$ be the character of an irreducible representation $V$ of $G$, and $g ∈ G$. Then a classical bound on the trace of $g$ is given by: $|χ(g)|² ≤ |C_G(g)|$, where $C_G(g)$ ...
• 2,273
273 views

### Embedding rank of finite groups and quotients

Let $G$ be a finite group, and $n$ a positive integer. It is not hard to check that the following are equivalent: For every $g\in G\setminus\{1\}$ there is a subgroup $H\leq G$ with $|G/H|\leq n$ ...
• 55.6k
70 views

### Stem extensions and quotients of Schur covers

Suppose that $G$ is a finite group, and that $\Gamma$ is a central extension of $G$ by $A$, that is $$1 \rightarrow A \rightarrow \Gamma \rightarrow G \rightarrow 1$$ with the image of $A$ contained ...
• 1,300
302 views

### A projectivity property in the category of groups

Let $F_r$ be the free group on $r$ generators, let $G$ and $H$ be finite groups, and let $F_r\xrightarrow{\alpha}H\xleftarrow{\beta}G$ be surjective homomorphisms. It is then easy to see that we can ...
• 55.6k
252 views

### Automorphisms of powers of finite simple groups

It is a theorem that a finite nontrivial group $G$ has no proper nontrivial characteristic subgroups if and only if $G \cong S^n$ where $S$ is simple and $n > 0$ is the number of copies of $S$ in a ...
• 355
1 vote
99 views

### Generators of a Coxeter group

Let $(W,S)$ be a Coxeter system and assume that $|S|$ is finite. Certainly, $W$ is generated by $|S|$ simple reflections. My question is: Can $W$ be generated by fewer reflections? (Including non-...
680 views

### Are there any non-conjugation "extendible automorphisms" in the category of finite groups?

Let $\mathbf{Grp}$ be the category of groups. Given a subcategory $\mathscr{G}$ of $\mathbf{Grp}$ and $G\in\mathit{Ob}(\mathscr{G})$, a $\mathscr{G}$-extendible map on $G$ will here mean an assignment ...
• 20.7k
1 vote
159 views

### Isomorphism classes of finite $\mathbb{N}$-groups

Where can I find resources on isomorphism classes of finite $\mathbb{N}$-groups, i.e. groups acted on by the monoid $(\mathbb{N}, +)$? I edited this question to be more focused on what I'm interested ...
• 355
294 views

### What is the minimal genus of a surface acted on by the symmetric group $S_n$?

For $G$ a finite group, it is easy to construct a (connected, orientable) surface with a faithful action of $G$. E.g.: take a disjoint union of $G$ many spheres, and add a 1-handle for every edge in ...
65 views

### Upper bound for Davenport constant $D(S_n)$

I am working on the Davenport constant $D(G)$ for $S_n$ and $A_n$, i.e., the minimal number $d$ such that every sequence (multiset) of $d$ elements contains some subsequence giving identity as a ...
97 views

### Cohomological characterization of when $f: \pi_1(\Sigma_g) \to P$ factors through $F_g$ when $P$ is perfect

In previous questions on this site such as this one, it has been asked when a map $\varphi \colon G \to H$ of finitely generated groups factors through a free quotient meaning that there exists a ...
1 vote
70 views

### Relationship between units and primitive characters 2

This is a follow up to this question. Let $(R,+,\cdot)$ be a finite ring. Definition Given the dual group $\widehat{R}$ of $(R,+)$, a character $\chi\in\widehat{R}$ is said to be primitive with ...
1 vote
254 views

• 217
539 views

### Groups whose derived length is logarithmic in the order?

Is there a class of solvable groups $G$ having a derived length $O(\log\lvert G\rvert)$? See Wikipedia for the definition of Big-Oh ($O$) and the definition of derived series of a group. Any help ...
• 217
157 views

### Largest primitive subgroup of $\mathrm{GL}_8(\mathbb{C})$ of order $2^a 3^b 5^c$

The paper "Bounds for finite primitive complex linear groups" by M. Collins computes the largest possible value of $[G:Z(G)]$ for $G$ a primitive subgroup of $\mathrm{GL}_N(\mathbb{C})$, for ...
• 123
392 views

### Number of finite groups: is $\operatorname{gnu}(4n) \geq 2 \operatorname{gnu}(n)$?

Let $\operatorname{gnu}(n)$ be the number of finite groups of order $n$. Question: Is it true that $\operatorname{gnu}(4n) \geq 2 \operatorname{gnu}(n)$ for all $n \geq 1$? Surely this must be true, ...
• 439
141 views

### What is the finite group $(\operatorname {PCO}^{\circ}_{2n})^{+}(q)$

In Table 22.1 on Page 193 of Malle & Testerman's book "Linear algebraic groups and finite groups of Lie type", the fixed point subgroup $G^F$ (where $F$ is a Steinberg endomorphism) of ...
• 165