# Questions tagged [group-theory]

A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory is the study of groups.

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### Is there a characterization of groups with the property $\forall N\unlhd G,\:\exists H\leq G\text{ s.t. }H\cong G/N$?

A common mistake for beginning group theory students is the belief that a quotient of a group $G$ is necessarily isomorphic to a subgroup of $G$. Is there a characterization of the groups in which ...
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### Geometric way to view the truncated braid groups?

This is perhaps a vague question, but hopefully there exists literature on the subject. The question is motivated by an answer I gave to this question. I also asked a related question on MO, although ...
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### A question about Sylow subgroups and $C_G(x)$

Let $G=PQ$ where $P$ and $Q$ are $p$- and $q$-Sylow subgroups of $G$ respectively. In addition, suppose that $P\unlhd G$, $Q\ntrianglelefteq G$, $C_G(P)=Z(G)$ and $C_G(Q)\neq Z(G)$, where $Z(G)$ is ...
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### Subgroups as isotropy subgroups and regular orbits on tuples

Is there some natural or character-theoretic description of the minimum value of d such that G has a regular orbit on Ωd, where G is a finite group acting faithfully on a set Ω? Motivation: In some ...
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### Can Erdős-Turán $\frac{5}{8}$ theorem be generalised that way?

Suppose for an arbitrary group word $w$ ower the alphabet of $n$ symbols $\mathfrak{U_w}$ is a variety of all groups $G$, that satisfy an identity $\forall a_1, … , a_n \in G$ $w(a_1, … , a_n) = e$. ...
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### What is $\tau(A_n)$?

Suppose G is a finite group. Define $\tau(G)$ as the minimal number, such that $\forall X \subset G$ if $|X| > \tau(G)$, then $XXX = \langle X \rangle$. What is $\tau(A_n)$? Similar problems for ...
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### Is there a “ping-pong lemma proof” that $\langle x \mapsto x+1,x \mapsto x^3 \rangle$ is a free group of rank 2?

Let $f,g\colon \mathbb R \to \mathbb R$ be the permutations defined by $f\colon x \mapsto x+1$ and $g\colon x \mapsto x^3$, or maybe even have $g\colon x \mapsto x^p$, $p$ an odd prime. In the book, ...
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### Iwahori versus Bruhat decompositions

I am faced with the following issue that I do not understand but seems contradictory, coming from the book of Roberts and Schmidt about $GSp(4)$. Consider a local non-archimedean field $F$, let $p$ be ...
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### Proving there is no non-abelian finite simple group of order a Fibonacci number

"Prove there does not exist a finite simple non-abelian group of order of a Fibonacci number" I would like to answer the above question, but I currently have few ideas of where to begin. I ...
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### How big must the union of a group's Sylow p-subgroups be?

For various orders $n$ it's a common exercise to prove that a finite group $G$ of order $n$ can't be simple by using the Sylow theorems to show that there is some prime $p \mid n$ such that the number ...
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### Maximal subgroups that force solvability.

For which finite groups $M$ is it the case that every finite group $G$ with $M$ as a maximal subgroup solvable? If $M$ satisfies this condition then $M$ is solvable. Also, if $M$ is abelian then $M$ ...
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### Groups of order $180$, $540$, $1080$ are not simple.

Here's how I solve the problems. Thanks for pointing out what might be the weakness of my solutions. Actually, what I want are other ways of solving this kind of problems, appart from counting the ...
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### If $g$ is commutator then so is $g^m$ for $(m,o(g))=1$

There are certain theorems in finite group theory whose proofs involve character theory and for which there are still no character-free proofs. Among such is Frobenius theorem on transitive ...
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### Abstract proof that $\lvert H^2(G,A)\rvert$ counts group extensions.

$\DeclareMathOperator{\Hom}{Hom}$ $\DeclareMathOperator{\im}{im}$ $\DeclareMathOperator{\id}{id}$ $\DeclareMathOperator{\ext}{Ext}$ $\newcommand{\Z}{\mathbb{Z}}$ Let $G$ be a group, let $A$ be a $G$-...
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### The multiplicative group of all complex $2^n$-th roots of unity, where $n = 0 , 1, 2 , \ldots$

Let $G$ be the multiplicative group of all complex $2^n$-th roots of unity, where $n = 0 , 1, 2 , \ldots$. Then assess the following claims: Every proper subgroup of $G$ is finite. $G$ has a ...
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### The division algebras arising in the Wedderburn decomposition of a finite group modulo its radical in characteristic $p$

The following question is probably straightforward for those who know. However, I am used to working either over splitting fields or in characteristic zero. Question. Let $G$ be a finite ...
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### Higman group with 3 generators is trivial

The group $G$ generated by $x,y,z$ subject to the relations $[x,y]=y$, $[y,z]=z$, $[z,x]=x$ is trivial. This isn't the case for the corresponding group with 4 generators, which is the famous Higman ...
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### Seeing symmetries

Preliminaries Let $[n] = \{0,\dots,n-1\}$ and $P([n])$ be the power set of $[n]$. Let the correlation between two subsets $x,y$ of $[n]$ be the number $\kappa(x,y) = 1 - \frac{2}{n}|x\triangle y|$ ...
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### Developing Intuition for the Monster Group

Here is my context. I recently finished my undergraduate studies and am moving on to graduate work. I have learned up to and including the Sylow theorems, group actions, and conjugacy classes. Over ...
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### Prove that the group $\mathrm{GL}(n, \mathbb{Z})$ is finitely generated

Knowing that for $n \geq 2$, $\mathrm{GL}(n, \mathbb{Z}) = \big\{ A \in \mathrm{M}_{n,n}(\mathbb{Z}) \mid \det(A) \in \{ 1, −1 \} \big\}$ is a group with respect to matrix multiplication, prove that ...