# 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.

9,012 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
1k views

### 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 $\forall a_1, … , a_n \in G$ $w(a_1, … , a_n) = e$. ...
• 15.4k
2k views

### 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

• 369
457 views

### 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 ...
• 3,188
545 views

### 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 $G$-...
• 4,166
612 views

### 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 $M$ ...
• 4,166
380 views

### 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 ...
• 1,315
249 views

### 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. ...
• 2,743
471 views

### 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 $\langle H, H' \rangle \in S$. ...
• 625
164 views

### 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 $G$ ...
• 151k
313 views

### 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 ...
• 15.4k
349 views

• 2,507
3k views

### 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 ...
• 2,985
411 views

### 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 ...
• 171
766 views

• 1,301
219 views

• 15.4k
92 views

### 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 ...
• 895
327 views

### Why do we trust the Classification of Finite Simple Groups?

It seems to me there are a two main reasons to believe a theorem/conjecture to be true: Because it has a correct proof (e.g. the Feit-Thompson Theorem, Dirichlet's Theorem) Because there is an ...
• 4,074