Skip to main content

Questions tagged [finite-groups]

Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

2,387 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
84 votes
0 answers
2k 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 $\forall a_1, … , a_n \in G$ $w(a_1, … , a_n) = e$. ...
Chain Markov's user avatar
  • 15.6k
74 votes
0 answers
1k views

If $f(n)$ is the number of groups of order $n$, then is $f(a)\cdot f(b)\leq f(a\cdot b)$?

Let $f(n)$ be the number of groups of order $n$ up to isomorphism. We want to prove that: $$f(a) \cdot f(b) \leq f(a \cdot b)$$ for all nonnegative integers $a$ and $b$. Our progress: If $a \cdot b \...
Jorge Rael's user avatar
27 votes
0 answers
2k views

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 ...
Sidharth Ghoshal's user avatar
23 votes
0 answers
623 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 $M$ ...
Thomas Browning's user avatar
19 votes
0 answers
475 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 $\langle H, H' \rangle \in S$. ...
kevkev1695's user avatar
18 votes
0 answers
322 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 ...
Chain Markov's user avatar
  • 15.6k
18 votes
0 answers
916 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 ...
Noah Schweber's user avatar
17 votes
0 answers
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 ...
frafour's user avatar
  • 3,025
17 votes
0 answers
429 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 ...
Alex's user avatar
  • 171
17 votes
0 answers
774 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}\...
Alexander Gruber's user avatar
  • 27.1k
16 votes
0 answers
3k views

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: ...
user197284's user avatar
15 votes
0 answers
512 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 ...
Benjamin Steinberg's user avatar
14 votes
0 answers
434 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 $\...
Aranya Lahiri's user avatar
14 votes
0 answers
221 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 $...
KcH's user avatar
  • 738
13 votes
0 answers
196 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 ...
Lê Thành Dũng 'Tito' Nguyễn's user avatar
12 votes
0 answers
397 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 ...
Zoe Allen's user avatar
  • 4,493
12 votes
0 answers
511 views

How do I know if an irreducible representation is a permutation representation?

I have a vague question, a less vague question and a lot of vaguer questions about permutation representations of a finite group $G$. Vague question. Recall that if $G$ acts on a finite set $X$, we ...
PseudoNeo's user avatar
  • 9,769
12 votes
0 answers
93 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 ...
fulges's user avatar
  • 915
11 votes
0 answers
209 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 $A$...
Chain Markov's user avatar
  • 15.6k
11 votes
0 answers
120 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,\...
Jack Schmidt's user avatar
  • 55.7k
10 votes
0 answers
170 views

A necessary condition for non-vanishing of irreducible characters of $S_n$

The following is Corollary 2.4.9 from the book "The Representation theory of the Symmetric Group" by James and Kerber. For partition $\alpha,\beta \vdash n$, $\chi_{\alpha}(\beta)\neq 0 \...
Riju's user avatar
  • 4,095
10 votes
0 answers
148 views

Is the matrix $\sum_{g \in G} a_g \rho(g)$ normal and what further properties does it have?

Let $\rho$ be the regular representation of $G$. $S \subset G$ a generating set, $|g| := |g|_S=$ word length with respect to $S$. Then I construct such a matrix, where we have some ordering $g_i$ of ...
user avatar
10 votes
0 answers
240 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 ...
JP McCarthy's user avatar
  • 8,439
10 votes
0 answers
405 views

Slick proof that Coxeter groups $H_3$ and $H_4$ are finite?

I'm planning to teach a course on reflection/Coxeter groups next Fall, and have started outlining the first days, which will presumably be the classification of finite reflection groups. Here is an ...
David E Speyer's user avatar
10 votes
0 answers
186 views

Restricting irreps of $S_n$ to $D_n$ of order $2 n$

I would like to know how to restrict the irreps of the symmetric group $S_n$ to the dihedral group $D_n$ of order $2 n$. We know that $D_n < S_n$. Symmetric group $S_n$ Due to Hardy and Ramanujan ...
Omar Shehab's user avatar
9 votes
0 answers
210 views

Why is studying centralizers the/a key to classifying finite groups?

In this MO thread https://mathoverflow.net/questions/38161/heuristic-argument-that-finite-simple-groups-ought-to-be-classifiable, Borcherds says One problem, as least with the current methods of ...
D.R.'s user avatar
  • 8,805
9 votes
0 answers
316 views

Inverse Galois problem

The inverse Galois problem conjectures that every finite group is (isomorphic to) the Galois group of some Galois extension of $\mathbb Q$, however it is not known. My question is: what is the ...
Régis's user avatar
  • 843
9 votes
1 answer
272 views

Complexity of testing if a binary operation is a group

Given a binary operation specified as an $n \times n$ Cayley table, what is the complexity of the best deterministic algorithm for testing if the binary operation is a group? There's a fairly simple ...
Qudit's user avatar
  • 3,231
9 votes
1 answer
330 views

Reference request: groups of order $p^4$.

I am looking for a textbook or a paper which include the classification of groups of order $p^4$ ($p$ is prime) using generators and relations. In particular I like to understand which group $G$ "...
Ofir Schnabel's user avatar
9 votes
0 answers
106 views

Missing $\{2,p\}$-Hall subgroups in finite non-abelian simple groups

Can anybody tell me how to prove this theorem? Theorem: Suppose that $G$ is a finite non-abelian simple group. Then there exists an odd prime $p\in\pi(G)$ such that $G$ has no $\{2,p\}$-Hall ...
user94782's user avatar
  • 241
8 votes
0 answers
146 views

an interesting coincidence with group orders: is there some theorem explaining it?

The sequence OEIS A333646: $1,6,15,28,30,33,40,42,51,66,69,84,91,95,102,105,117,120,135,138,140,141,145,159,165,182,186,190,210,213,\cdots$ Contains numbers that are divisible by the largest prime ...
Michał Zapała's user avatar
8 votes
0 answers
188 views

When does a finite group $G$ only have a single maximal subgroup containing given $A<G$?

Let $G$ be a finite group. Let $A<G$. When is it true that $G$ has only a single maximal subgroup containing (or equal to) $A$? The necessary condition is that there must be $x \in G$ such that $\...
Michał Zapała's user avatar
8 votes
0 answers
190 views

Automorphisms for direct products of finite commutative nilpotent rings.

Let $(R, +, \cdot)$ be an associative commutative nilpotent ring of cardinality $2^n$ such that $$ r^2 = 0, $$ for every $r\in R$. Also $(V, +)$ is a vector space over $\mathbb{F}_2$. Let $\...
Mikhail Goltvanitsa's user avatar
8 votes
0 answers
96 views

What number of group elements of a specific order are (non-)realisable?

My question is as follows: Let $G$ be an arbitrary finite group. Let $n_r(G) \in \mathbb{N}_0$ be the amount of elements that have exactly order $r \geq 2$ in $G$. For which numbers $k \in \mathbb{N}...
TheOutZ's user avatar
  • 1,256
8 votes
0 answers
131 views

Primitive permutation groups containing a cycle

I am trying to prove the following result: Let $G$ be a primitive permutation group on $\Omega$ of degree $n$ that contains a cycle $g$ fixing $k \geq 3$ points. Then, $A_n \leq G$ where $A_n$ is ...
TissuePaper's user avatar
8 votes
0 answers
178 views

Categorifying $1^2+2^2+3^2+\cdots+24^2=70^2$

Does $1^2+2^2+3^2+\cdots+24^2=70^2$ or a simple equivalent have an interesting categorification? Lots of combinatorical identities do. For instance, the Vandermonde convolution identity and the ...
anon's user avatar
  • 152k
8 votes
0 answers
107 views

Existence of primitive permutation group one of whose arc stabilisers is normal in the point stabiliser

Let $G$ be a primitive permutation group with point stabiliser $G_\alpha$ for some $\alpha$. For $\beta\ne\alpha$, by an arc stabiliser we mean $G_{\alpha\beta}=G_\alpha\cap G_\beta$ and an edge ...
Groups's user avatar
  • 2,876
8 votes
1 answer
231 views

A simple group with $|\operatorname{Syl}_p⁡ G| \le 6$ is cyclic

Let $G$ be a simple, finite group, s.t. for every prime $p$, it satisfies $k_p=\left|\operatorname{Syl}_p⁡ G\right| \le 6$. Show that $G$ is cyclic. My attempt: Let $n=p_1^{e_1}p_2^{e_2}\ldots p_r^{...
Roy Sht's user avatar
  • 1,349
8 votes
1 answer
357 views

Let $G$ be a nilpotent group generated by a finite set of torsion (i.e. finite order) elements. Show that $G$ is finite.

Given: Let $G$ be a nilpotent group generated by a finite set of torsion (i.e. finite order) elements. Show that $G$ is finite. Also would love to know if it's possible to show that an infinite ...
Ilan Aizelman WS's user avatar
8 votes
0 answers
442 views

Proof of a Burnside theorem without character theory?

Burnside proved, by use of character theory, that if a finite group $G$ has a conjugacy class $C$ such that $\vert C \vert$ is a prime power $> 1$, then $G$ is not simple. Let us call this ...
Panurge's user avatar
  • 1,695
8 votes
0 answers
138 views

An upper bound on automorphism orbit lengths in nonabelian finite simple groups

In what follows, for a group $G$, $\operatorname{Aut}(G)$ and $\operatorname{Out}(G)$ denote the automorphism and outer automorphism group of $G$ respectively, and for $x\in G$, $x^G$ and $x^{\...
Alexander Bors's user avatar
8 votes
0 answers
439 views

Reading the centralizer off of the character table

Assume that I am given the table of irreducible characters of a finite group $G$. I realize that we can see the order of the centralizer of any element $g \in G$ by summing the squares in the ...
user392576's user avatar
8 votes
0 answers
221 views

Is the group $G_{n,\phi} = \langle x_1 , \dots, x_n \mid x_i^2, (x_i x_j)^4, x_i x_{\phi(i)} x_{i+1} x_{\phi(i)} \rangle$ abelian?

I am working on a family of finitely presented groups and I asking me the following question. Let $\phi$ be an application from $\left\{1,\dots,n\right\}$ to $\left\{1,\dots,n\right\}$ (not necessary ...
Benoît Guerville-Ballé's user avatar
8 votes
0 answers
277 views

$|G:H|=p^n$ means $O_p(H)\leq O_p(G)$?

Let $H\leq G$ (finite group) and $|G:H|=p^n$, ($p$ is a prime number) prove that: $$O_p(H)\leq O_p(G)$$ note: $O_p(G)$ defined as the intersection of all Sylow-$p$ groups in $G$ I try to prove $G_p\...
chenyuandong's user avatar
8 votes
0 answers
609 views

Classify groups of order 171

This is a problem from Stanford Algebra Qualifying Exam, Fall 1998. I know the standard way is to use Sylow theorems and semidirect product. $171 = 9\cdot 19$. By Sylow theorems, $n_3|19$ and $n_3\...
Zhulin Li's user avatar
  • 605
8 votes
0 answers
168 views

A connection between nonplanar complete graphs and the alternating group?

I went to an undergrad's senior honors thesis presentation a few days ago. She was discussing crossing numbers and mentioned that complete graphs $K_n$ are nonplanar iff $n \geq 5$. ?Coincidentally? $...
Bill Cook's user avatar
  • 29.4k
8 votes
0 answers
492 views

Historical Question about Schur-Zassenhaus Theorem

I couldn't find any historical information about Schur-Zassenhaus theorem in many books or even papers which mention this theorem. I think, Schur proved that if $G$ is a finite group and if $N$ is ...
Beginner's user avatar
  • 10.9k
8 votes
0 answers
3k views

Discrete subgroups of SU(n) and SO(n).

Thank you very much for your concern. I am in physics background, any simpler but complete explanation would be helpful. I would like to know whether there is a complete understanding of discrete ...
miss-tery's user avatar
  • 1,103
8 votes
0 answers
136 views

Fixed points of coset operation

Let $G$ be a finite group which operates on two finite sets $E_1$ and $E_2$. Say that $E_1$ and $E_2$ are weakly $G$-isomorphic if for every $g \in G$, $\mathrm{Card}(E_1^g)=\mathrm{Card}(E_2^g)$, ...
timofei's user avatar
  • 81
8 votes
0 answers
521 views

a split exact sequence of finite groups

Suppose G has a cyclic normal subgroup $\langle a\rangle$ of order $m$ and prime power index $s$ such that $m$ and $s$ are relatively prime. Then the following exact sequence splits: $$1 \...
Yubin's user avatar
  • 247

1
2 3 4 5
48