# Tagged Questions

Questions about the branch of abstract algebra that deals with groups.

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### Does $\mathrm{Aut}(\mathrm{Aut}(…\mathrm{Aut}(G)…))$ stabilize?

Purely for fun, I was playing around with iteratively applying $\DeclareMathOperator{\Aut}{Aut}\Aut$ to a group $G$; that is, studying groups of the form $${\Aut}^n(G):= \Aut(\Aut(...\Aut(G)...))$$ ...
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### Cool problems to impress students with group theory [closed]

Since this forum is densely populated with algebraists, I think I'll ask it here. I'm teaching intermediate level algebra this semester and I'd like to entertain my students with some clever ...
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### Does every non-empty set admit a group structure (in ZF)?

It is easy to see that in ZFC, any non-empty set $S$ admits a group structure: for finite $S$ identify $S$ with a cyclic group, and for infinite $S$, the set of finite subsets of $S$ with the binary ...
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### $A$ is isomorphic to $A \oplus \mathbb{Z}^2$, but not to $A \oplus \mathbb{Z}$

Are there abelian groups $A$ with $A \cong A \oplus \mathbb{Z}^2$ and $A \not\cong A \oplus \mathbb{Z}$?
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### Why do Groups and Abelian Groups feel so different?

Groups are naturally "the symmetries of an object". To me, the group axioms are just a way of codifying what the symmetries of an object can be so we can study it abstractly. However, this heuristic ...
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### How feasible is it to prove Kazhdan's property (T) by a computer?

Recently, I have proved that Kazhdan's property (T) is theoretically provable by computers (arXiv:1312.5431, explained below), but I'm quite lame with computers and have no idea what they actually ...
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### Non-amenable groups with arbitrarily large Tarski number?

Just out of curiosity, I wonder whether there are non-amenable groups with arbitrarily large Tarski numbers. The Tarski number $\tau(G)$ of a discrete group $G$ is the smallest $n$ such that $G$ ...
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### Can we ascertain that there exist an epimorphism $G\rightarrow H?$

Let $G,H$ be finite groups. Suppose we have a epimorphism $G\times G\rightarrow H\times H$. Can we find an epimorphism $G\rightarrow H$? A fellow graduate student asked me this question during TA ...
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### 5/8 bound in group theory

The odds of two random elements of a group commuting is the number of conjugacy classes of the group $$\frac{ \{ (g,h): ghg^{-1}h^{-1} = 1 \} }{ |G|^2} = \frac{c(G)}{|G|}$$ If this number exceeds ...
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### Order of an automorphism of a finite group

Let G be a finite group of order n. Must every automorphism of G have order less than n? (David Speyer: I got this question from you long ago, but I don't know whether you knew the answer. I stil ...
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### Transitivity on $\mathbb{N}_0$ — a 42 problem

Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition $\tau_{r_1(m_1),r_2(m_2)}$ be the ...
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### Normalizers in symmetric groups

Question: Let $G$ be a finite group. Is it true that there is a subgroup $U$ inside some symmetric group $S_n$, such that $N(U)/U$ is isomorphic to $G$? Here $N(U)$ is the normalizer of $U$ in $S_n$. ...
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### Does $SL_3(R)$ embed in $SL_2(R)$?

Is there any non-trivial ring such that $SL_{3}(R)$ is isomorphic to a subgroup of $SL_{2}(R)$? $SL_{3}(\mathbb{Z})$ is not an amalgam, and has the wrong number of order $2$ elements to be a subgroup ...
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### 1 rectangle <= 4 squares

Almost 25 years ago a professor at Indiana U showed me the following problem: given a map $\mathbb{Z}^2\rightarrow\mathbb{R}$ such that the sum inside every square (parallel to the axes) is $\leq1$ ...
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### Isomorphic general linear groups implies isomorphic fields?

Suppose $n > 1$ is a natural number. Suppose $K$ and $L$ are fields such that the general linear groups of degree $n$ over them are isomorphic, i.e., $GL(n,K) \cong GL(n,L)$ as groups. Is it ...
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### What is the current status of the Kaplansky zero-divisor conjecture for group rings?

Let $K$ be a field and $G$ a group. The so called zero-divisor conjecture for group rings asserts that the group ring $K[G]$ is a domain if and only if $G$ is a torsion-free group. A couple of good ...
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### Does a finite simple group of order divisible by $60$ have $A_{5}$ as a subgroup?

The title says it all really. I know that in some "small" simple groups such as ${\rm PSL}(2,p)$ it is known that if the order is divisible by $60,$ then $A_{5}$ does occur as a subgroup ( this is ...
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### How much of character theory can be done without Schur's lemma or the Artin-Wedderburn theorem?

This is a somewhat imprecise question, as I am not sure how exactly how to formalise how to do mathematics "without" a certain key tool, but hopefully the intent of the question will still be clear. ...
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### Fantastic properties of Z/2Z

Recently I gave a lecture to master's students about some nice properties of the group with two elements $\mathbb{Z}/2\mathbb{Z}$. Typically, I wanted to present simple, natural situations where the ...
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### What algebraic group does Tannaka-Krein reconstruct when fed the category of modules of a non-algebraic Lie algebra?

Let $\mathfrak g$ be a finite-dimensional Lie algebra over $\mathbb C$, and let $\mathfrak g \text{-rep}$ be its category of finite-dimensional modules. Then $\mathfrak g\text{-rep}$ comes equipped ...
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### How much of the ATLAS of finite groups is independently checked and/or computer verified?

In a recent talk Serre made some comments about proofs that rely on the classification of finite simple groups (CFSG) and on the ATLAS of Finite Groups. Namely, he said that a proof that relied on the ...
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### Infinitely many solutions of a diophantine equation

If $P(x,y,...,z)$ is a polynomial with integer coefficients then every integer solution of $P=0$ corresponds to a homomorphism from $\mathbb{Z}[x,y,...,z]/(P)$ to $\mathbb{Z}$. So there are infinitely ...
Update: The negative answer to the following question has been provided by Matthew Daws, who won, but also rejected, the bounty of 100 euro that I set over the question. Let $\mathcal M(\mathbb Z)$ ...
In his answer to this previous MO question, Gjergji Zaimi referred to the statement that for every finite group $G$ of order $n$, there is a bijection $\sigma \colon G \to \mathbb{Z}/n\mathbb{Z}$ ...