# Questions tagged [group-theory]

For questions about the study of algebraic structures consisting of a set of elements together with a well-defined binary operation that satisfies three conditions: associativity, identity and invertibility.

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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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### 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, ... 1k views

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### Is there a group $G$ for which $\mathrm{Aut}(G) \simeq (\mathbb{R},+)$?

I know the classic theorem that $(\mathbb{Q},+)$ cannot be expressed as an automorphism group, i.e. there is no group $G$ such that $\mathrm{Aut}(G)\simeq (\mathbb{Q},+)$. Theorem A. If $L$ is a ...
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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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### 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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### Permutations of Rubik's cube such that no adjacent sticker is the same

I've always wondered, what is the number of possible permutations of the Rubik's cube such that any two adjacent stickers has a different color. By a permutation I mean a configuration of the cube ...
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### Does every finitely generated group have finitely many retracts up to isomorphism?

The infinite dihedral group $D_\infty = \langle a,b \mid a^2 = b^2 = \text{Id}\rangle$ is a finitely generated group with infinitely many cyclic subgroups of order 2, every one of which is a retract. ...
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### Are groups with this property already studied?

Let $\Omega$ be a finite group, let $G$ be a subgroup of $\Omega$ and let $S$ be a set of subgroups of $\Omega$ such that for $H, H'\in S$ we have $H\cap H' \in S$ and $\langle H, H' \rangle \in S$. ...
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### How do permutations of $\Bbb N$ affect series?

Let $G$ be the group of all permutations of $\mathbb{N}$ and $\sum a_n$ a conditionally convergent series of reals. What do we know about how $G$ "acts" on this series? We can partition $G$ ...
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### Probability of a group being finite

Suppose $F_m := F[x_1, … , x_m]$ is a free group on $m$ generators $x_1, … , x_m$ and lets define Cayley ball $B_m^n := \{e, x_1, x_1^{-1}, … , x_m, x_m^{-1}\}^n$ as the set of all elements with ...
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### Was Atiyah's proof of the odd order (Feit-Thompson) theorem false?

I read last year that Atiyah thought he had found a proof of the odd order theorem of only 12 pages, using $K$-theory, and that people were trying to figure out if it was correct or not. But I never ...
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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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### Small group characterizing identity matrix

I am looking for a small (say, finite and of small cardinality) subgroup of the general linear group whose centralizer consists only of scalar matrices. I work over complex numbers. A more precise ...