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
81
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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 $\forall a_1, … , a_n \in G$ $w(a_1, … , a_n) = e$. ...
- 15.4k
73
votes
0
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2k
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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, ...
user29123
35
votes
0
answers
1k
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Can you make the triviality of $\langle a,b,c \mid aba^{-1} = b^2, bcb^{-1} = c^2, cac^{-1} = a^2 \rangle$ more trivial?
I recently learned the following pleasant fact. (It was in the proof of Proposition 3.1 of this paper - but don't worry, there's no model theory in this question.)
Let $G$ be a group, and let $a,b,c\...
- 73.8k
27
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0
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440
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Wave equation: predicting geometric dispersion with group theory
Context
The wave equation
$$
\partial_{tt}\psi=v^2\nabla^2 \psi
$$
describes waves that travel with frequency-independent speed $v$, ie. the waves are dispersionless. The character of solutions is ...
- 4,506
27
votes
0
answers
1k
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Is there a group theoretic proof that $(\mathbf Z/(p))^\times$ is cyclic?
Theorem: The group $(\mathbf Z/(p))^\times$ is cyclic for any prime $p$.
Most proofs make use of the fact that for $r\geq 1$, there are at most $r$ solutions to the equation $x^r=1$ in $\mathbf Z/(p)$...
- 4,760
27
votes
0
answers
2k
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Lowest dimensional faithful representation of a finite group
How does one compute the lowest dimensional faithful representation of a finite group?
This question originated in the context of given a finite group $G$: trying to find the lowest dimensional shape ...
- 16.6k
26
votes
0
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1k
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Description of Levi factors and unipotent radicals of parabolic subgroups in classical groups
For an algebraic group $G$ over an algebraically closed field $k$, a parabolic subgroup $P$ has factorization $P = Q \rtimes L$, where $Q$ is the unipotent radical of $P$ and $L$ is some Levi factor ...
- 11.8k
26
votes
0
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386
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$f\colon I\rightarrow G$ and Gromov $\delta$-hyperbolicity
Please recall that $\left|\int_0^1 f(t)\,dt -w\right|\leq \int_0^1|f(t)-w|\,dt$. In general, let $(X,d)$ be a metric space. Given a function $f:I\to X$ let $m_f\in X$ be such that $d(m_f,w)\leq \int_0^...
- 369
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0
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457
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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 ...
- 3,188
24
votes
0
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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
23
votes
0
answers
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
20
votes
1
answer
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
19
votes
0
answers
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
19
votes
0
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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
18
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0
answers
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
18
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0
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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
18
votes
0
answers
349
views
Can the set of endomorphisms of $(\mathbb{R},+)$ have cardinality strictly between $\mathfrak c^ c$ and $\frak c $?
Let $\mathfrak{c}$ be the cardinality of the continuum. It is well known that in ZFC that the reals under addition has $\frak{ c^c}$(cardinality of maps to itself, and in ZFC this is the same as $2^\...
user29123
18
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0
answers
224
views
Have "groupy" numbers been studied before?
In number theory, a positive integer $n$ is called highly composite if it has more divisors than any smaller positive integer. This notion has been studied by several notable mathematicians; for ...
- 12.2k
18
votes
0
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895
views
What groups are semidirect products of simple groups?
The question I want to ask is actually slightly broader than that in the title: what is the smallest class of finite groups which contains all finite simple groups, and is closed under semidirect ...
- 240k
17
votes
0
answers
330
views
Cardinal numbers of right factors of a group
Let $A$ and $B$ be subsets of a group $G$. The product $AB$ is called direct (and we denote it by $A \cdot B$, e.g., see this) if the representation of each element $x$ of $AB$ as $x=ab$, $a\in A$, $b\...
- 2,507
17
votes
0
answers
3k
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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 ...
- 2,985
17
votes
0
answers
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
17
votes
0
answers
766
views
Character theory of $2$-Frobenius groups.
Edit Summary: I've posted this on MO and received a partial answer there. Can anybody help me expand on this?
Definition. Let $G$ be a finite group and $F_1=\text{Fit}\,G$ and $F_2/F=\text{Fit}\...
- 26.7k
16
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0
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306
views
+150
Can nonisomorphic groups have near-identical Cayley tables?
Nonisomorphic groups can have very similar multiplication (Cayley) tables. For example, the two groups
\begin{align*}
\mathbb{Z}/9\mathbb{Z}&=\{\overset{a}{0},\overset{b}{1},\overset{c}{2},\...
- 121
15
votes
0
answers
505
views
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 ...
15
votes
0
answers
718
views
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 ...
- 159k
15
votes
0
answers
3k
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Induced representation, Ind(Res(U))
I am reading a book of Fulton and Harris "Representation theory, a first course".
Now it's all about representation theory of finite groups, and there is one exercise, which I can't solve:
...
- 661
15
votes
1
answer
323
views
Attempt to prove that every subgroup is an equalizer
I'm trying to prove:
For any subgroup $H$ of $G$, there is a group $T$ and homomorphisms $f,g:G\to T$ such that $f(x)=g(x)$ iff $x\in H$.
My idea is to construct a group which contains two copies ...
- 151
15
votes
0
answers
2k
views
Fundamental group of a compact manifold
In an article I am currently reading, the author tells us that for compact (finite dimensional topological) manifolds X and finite groups $\Gamma$, the set $$\mathrm{Hom}(\pi_1X,\Gamma)/\Gamma$$ where ...
- 20k
14
votes
0
answers
421
views
Subgroup structure of $\mathrm{SL}(2, p^2)$ and its irreducible characters
I am taking a course in representation theory of finite groups, and somehow I ended up getting assigned to write a report on subgroup structure and irreducible characters of $\mathrm{SL}(2,7)$ and $\...
- 1,301
14
votes
0
answers
219
views
Official name(s) for a certain type of p-group
I'm implementing a class of groups into Sage (sagemath.org), a computer algebra system, and I'm wondering if it has any official names. I found it in Gorenstein's "Finite Groups." It is there called $...
- 738
13
votes
0
answers
193
views
Infinite family of finite groups without surjections
Is there an infinite family $\mathcal{F}$ of finite groups such that
their exponent is bounded, i.e. $\exists N \geq 1 : \forall x \in G \in \mathcal{F}, x^N = 1_G$;
there does not exist any ...
13
votes
0
answers
640
views
Showing ${\rm Aut}(Q_{2^n})\cong{\rm Hol}(\Bbb Z_{2^{n-1}})$ for $n>3$
This is part of Exercise 5.3.4 of Robinson's, "A Course in the Theory of Groups (Second Edition)". According to this search, it is new to MSE.
The second part is here:
Automorphism group of ...
- 43.6k
13
votes
0
answers
278
views
Are all verbal automorphisms inner power automorphisms?
Suppose $G$ is a group. $\DeclareMathOperator{\Wa}{Wa}\DeclareMathOperator{\Tame}{Tame}\DeclareMathOperator{\Aut}{Aut}$
Lets call $\phi \in \Aut(G)$ verbal automorphism iff $\exists n \in \mathbb{N}, ...
- 15.4k
12
votes
0
answers
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
11
votes
0
answers
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
11
votes
0
answers
187
views
A group product
If $G$ and $H$ are two groups, and
$\triangleright$ and $\triangleleft$ are a left action and a right action
of $H$ on $G$ by group automorphisms such that
$$h\triangleright(g\triangleleft h')=(h\...
11
votes
0
answers
575
views
Proof of $\operatorname{Aut}(C_q)\cong C_{q-1}$ by group action?
Let $G$ and $H$ be groups, and $\varphi\colon G\to\operatorname{Aut}(H)$ a homomorphism. Then, further than all the results valid for a general group action on a set, the following additional one ...
user943729
11
votes
0
answers
201
views
A generalization of Feit–Thompson conjecture, for square-free integers
Few weeks ago I wondered about if the following conjecture is in the literature or well if it is possible to find a counterexample. I evoke a generalization of a well-known conjecture, I mean the Feit–...
11
votes
0
answers
201
views
Does asymmetric fraction of finite groups tend to $0$?
Let’s define asymmetric fraction of a finite group $G$ as the number $af(G) = \frac{|\{(g, a) \in G \times Aut(G)| a(g) = g\}|}{|G||Aut(G)|}$. Equivalently it can be defined as $P(A(X) = X)$, where $A$...
- 15.4k
11
votes
0
answers
269
views
Is there a theory of "almost symmetry" generalizing group theory?
Apologies for the inescapably soft question.
Does there exist a theory that aims to develop tools analogous to those of group theory, except for the study of objects that are merely almost ...
- 20.6k
11
votes
0
answers
281
views
Is this specific group finite?
I have the following group presentation:
$G=\left\langle a,b,c\ |\ a^2,b^{11},c^2,(ab)^{4},(ab^2)^6,ab^2abab^{-1}abab^{-2}ab^2ab^{-1},(ac)^3,(bc)^2\right\rangle$
Is $G$ finite?
GAP's ...
- 3,351
11
votes
0
answers
119
views
Cover and avoidance properties of chief factors: can we avoid exactly the ones we want?
If $G$ is a finite solvable group with chief series $1 = G_0 \leq \ldots \leq G_n = G$ (so each $G_i \unlhd G$ and if $G_{i-1} < H < G_i$ then $H$ is not normal in $G$) and $I \subseteq \{ 1,\...
- 55.3k
11
votes
0
answers
832
views
How strong is the statement that Thompson F is amenable?
Justin Moore's proof turned out to have an error
I just attended Justin Moore's talk on this today. Since I am neither a group theorist nor a combinatorist, and is not familiar with ultrafilters I ...
- 19.5k
10
votes
0
answers
145
views
Does this series $L_{i+1}(G)=\{g\in G\mid \varphi(g)g^{-1}\in L_i(G), \forall\varphi\in\operatorname{Aut}(G)\}$ have a name?
From the comment to this other question of mine, I have learned that the series ($i=0,1,2,\dots$):
$$Z_{i+1}(G)=\{g\in G\mid \varphi(g)g^{-1}\in Z_i(G), \forall\varphi\in\operatorname{Inn}(G)\} \tag 1$...
user1007416
10
votes
0
answers
237
views
Raising to the power of $i$
We all know the usual exponentiation $a^i$ in the complex setting; one can ask is there such a map in a given group; more specifically is there a criterion in the general sense such that given an ...
- 3,366
10
votes
0
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138
views
Is there a categorification of "(virtually) solvable"?
If this question doesn't make sense or is otherwise poor quality, then I'm sorry.
Motivation:
As part of my research, I study virtually solvable (1) groups. These are goups that have a solvable ...
- 43.6k
10
votes
0
answers
235
views
Upper Bound Lemma implies the Ergodic Theorem for Random Walks on Groups?
Cross posted on Mathoverflow
Ergodic Theorem A random walk on a finite group $G$ driven by a probability $\nu\in M_p(G)$ is ergodic if $\operatorname{supp}(\nu)$ is not concentrated
on a proper ...
- 8,208
10
votes
0
answers
244
views
Finite group of "linear substitutions"
From what I can tell, a linear substitution is an operation on a set of variables $x_1,\ldots,x_n$ which sends them to a new set of variables $y_1,\ldots, y_n$ via a linear transformation
$$\vec{y} = ...
- 17.1k
10
votes
1
answer
856
views
Is there a name for the group of real matrices whose determinant is an element of $\pm 1$?
The group of matrices whose determinant is non-zero is called the "general linear group", and the group of matrices whose determinant is $1$ is called the "special linear group". In between these two ...
- 67.2k