Skip to main content
MathOverflow

Questions tagged [computational-group-theory]

Ask Question

The tag has no summary.

94 questions
Newest Active Bountied Unanswered
Filter by
Sorted by
Tagged with
7 votes
1 answer
121 views

Let $K = \mathbb{Q}(\zeta_m)$ be a cyclotomic field, where $\zeta_m = e^{2\pi i / m}$. Suppose we are given explicit inputs: A finite set of generators for a finite subgroup $G \subset U(n)$, whose ...
César Galindo's user avatar
  • 1,858
4 votes
1 answer
506 views

I'm coding a small application that looks for periodic solutions to the gravitational n-body problem. I'm trying to better understanding the symmetries of solutions, which is made up of the product of ...
G. Fougeron's user avatar
  • 297
14 votes
1 answer
2k views

According to our practical experiments and literature search - computer algebra system GAP cannot "solve" Rubik's cube 4x4x4 and higher. That means cannot decompose given random element of ...
Alexander Chervov's user avatar
  • 26.4k
4 votes
0 answers
186 views

Question 1: Consider the symmetric group $S_n$ and some set of permutations $p_i$. Given permutation $g$ - what is known about the algorithmic complexity to decompose $g$ into product of $p_i$ ...
Alexander Chervov's user avatar
  • 26.4k
4 votes
1 answer
370 views

(Link to SE duplicate: https://math.stackexchange.com/questions/4959071/are-group-theoretic-markov-properties-on-groups-with-decidable-word-problems) The Adian-Rabin theorem says that if a property of ...
Perry Bleiberg's user avatar
  • 191
3 votes
1 answer
207 views

$\DeclareMathOperator\GL{GL}$Suppose $G$ is a subgroup of $\GL(n,q)$ given by a list of generators. What is known about the complexity of the corresponding "membership problem", that is, the ...
Pierre's user avatar
  • 2,359
11 votes
1 answer
841 views

I’m studying the homology groups of arithmetic groups such as $SL(5,\mathbb{Z})$. I saw in the answer to this post that we can use GAP to compute some of the homology groups for $SL(3,\mathbb{Z})$. Is ...
Noah B's user avatar
  • 575
5 votes
1 answer
347 views

Let $G < S_n$ be a permutation group of degree $n$, $\mathcal{P(n)}$ denote the set of all partitions of $n$, and $c: G \rightarrow \mathcal{P}(n)$, where $c(g)$ is the partition given by the ...
Victor Miller's user avatar
  • 4,746
8 votes
1 answer
531 views

It is known that the isomorphism problem for finitely presented groups is in general undecidable. What are some classes of groups whose isomorphism problem is known to be solvable in polynomial time? (...
Mithrandir's user avatar
  • 283
4 votes
0 answers
193 views

I am wondering if there is a good method to write down a finite equational basis for a finite group. Especially I am wondering if there is a good method in following situations: We can write a group ...
Todor Antic's user avatar
  • 93
3 votes
1 answer
179 views

Let $G$ be a finite group with trivial action on $\mathbb C^\times$. And given a 2-cocycle $\alpha$ in its Schur multiplier group $H^2(G,\mathbb C^\times)$, as an explicit map from $G\times G\to \...
JKDASF's user avatar
  • 273
2 votes
0 answers
77 views

Victor Shoup in this paper has given a lower bound for discrete logarithm. The algorithms that I have come across use discrete logarithms (extended discrete logarithms) to compute a basis for a finite ...
Vasac's user avatar
  • 21
0 votes
0 answers
136 views

A finite group $G$ is called integral if there is a finite group $H$ such that $G\cong H'$. In Araujo, Cameron, Casolo, Matucci's paper, integrals of groups, they tried to solve a problem as following:...
Zhaochen Ding's user avatar
  • 1
8 votes
2 answers
507 views

For various types of groups, there exist catalogues of those groups of the particular type which are "small" in a certain sense. — For example: The GAP Small Groups Library catalogizes ...
Stefan Kohl's user avatar
  • 19.9k
13 votes
0 answers
632 views

In this MO question, user Martin Brandenburg asks about God's number for $n \times n \times n$-cubes for $n>3$. Here, God's number $g(n)$ was defined as the smallest number $m$ such that every ...
Max Lonysa Muller's user avatar
  • 5,065
15 votes
1 answer
865 views

In my research I came up with the following question: Question: Let $H_1$ and $H_2$ be finite abelian subgroups of $\mathsf{GL}(n,\mathbb{Z})$. Define $$ H_1'=\left\{\begin{pmatrix} I_m &0\\0&...
Alejandro Tolcachier's user avatar
  • 467
8 votes
1 answer
648 views

Suppose we have a finite group $G$ whose presentation or Cayley table is given. Is there an algorithm (at least theoretically - without considering computational complexity) to compute the Cayley ...
Cloud jr's user avatar
  • 89
0 votes
0 answers
353 views

Is there an algorithm to compute automorphism group of a finite group? GAP has a function to do this, but while perusing their GitHub repo, I could not find an implementation. I'm struggling to find ...
Jerry Halisberry's user avatar
  • 1
5 votes
0 answers
273 views

What are some tools -- either theoretical/by hand or algorithmic/by computer -- that are useful for doing computations in finitely presented groups? In my particular case, I'm working with a finitely ...
Ethan Dlugie's user avatar
  • 1,509
4 votes
2 answers
285 views

Suppose we are given a group $G$ in terms of generators $t_1, ..., t_n$ which are order 2 in $S_m$ (however we don't assume anything other than that these elements generate $G$ and have order 2). What ...
manzana's user avatar
  • 395
4 votes
1 answer
525 views

We can easily get the character table of $\mathrm{PSL}(2,q)$ for some fixed small prime power $q$, we can just do (for example): ...
Sebastien Palcoux's user avatar
  • 27.6k
4 votes
1 answer
312 views

Recall that the fundamental group of a closed Riemann surface of genus $h$ has the presentation $$\Pi_h= \langle a_1, \,b_1, \ldots, a_h,\, b_h \; | \; [a_1, \, b_1]\ldots [a_h, \, b_h]=1 \rangle.$$ ...
Francesco Polizzi's user avatar
  • 68.7k
1 vote
2 answers
474 views

The computation below (part 1) shows that if two finite groups of order at most $100$ have the same (ordered) list of conjugacy class sizes, then they also have the same (ordered) list of (irreducible)...
Sebastien Palcoux's user avatar
  • 27.6k
3 votes
1 answer
261 views

Let $G$ be a finite group and $k$ be a finite field (big enough) whith char$(k)=p$ and $p\mid |G|$. Let $M$ be a finitely generated $kG$-module. We denote the first syzygy of $M$ by $\Omega(M)$, i.e....
Bernhard Boehmler's user avatar
  • 1,795
4 votes
1 answer
416 views

I would like to find ($24\times 24$) matrices representing the various conjugacy classes of Conway's group $\mathrm{Co}_0$ acting on the Leech lattice in the usual coordinate system given by the MOG. ...
Gro-Tsen's user avatar
  • 38.8k
9 votes
2 answers
896 views

A group G is said to have a factorization if there exist proper subgroups $A$ and $B$ such that $G = AB = \{ ab \ | \ a \in A, b \in B \}$. The paper Factorisations of sporadic simple groups (...
Sebastien Palcoux's user avatar
  • 27.6k
1 vote
1 answer
569 views

Let call $n$ a sporadic number if the set of groups $G \neq A_n,S_n$ having a core-free maximal subgroup of index $n$ is non-empty and contains only sporadic simple groups. By GAP, the set of all the ...
Sebastien Palcoux's user avatar
  • 27.6k
3 votes
1 answer
247 views

The book The maximal factorizations of the finite simple groups and their automorphism groups (by Martin W. Liebeck, Cheryl E. Praeger and Jan Saxl) provides a classification of all the triples $(G,A,...
Sebastien Palcoux's user avatar
  • 27.6k
2 votes
0 answers
267 views

I asked the same question in MSE, but I didn't get any answer. So I decided to post it here, too. In Langlands' program, Satake correspondence gives a correspondence between unramified representation ...
Seewoo Lee's user avatar
  • 2,255
0 votes
0 answers
143 views

I'm given a graph $G$ (<1000 vertices, large automorphism group), and a large number (~10^6-10^10) of different colorings of said graph. I have two tasks. Calculate the canonical coloring. I can ...
J Bausch's user avatar
  • 101
29 votes
5 answers
5k views

There are several well known classes of groups for which the word problem, conjugacy etc. are solvable in polynomial time (hyperbolic, automatic). Then there are several classes of groups like ...
MSL's user avatar
  • 401
11 votes
0 answers
234 views

I suspect that many permutation puzzles can be solved in $O(n \log n)$ moves, which has led me to the following question/conjecture: Suppose that 1. $P_i$ for $i<k=O(1)$ are permutations on an $n$ ...
Dmytro Taranovsky's user avatar
  • 8,668
7 votes
2 answers
925 views

I am in my final year of my doctoral study in Mathematics, where my research topic is $p$-groups, specifically classification of $p$-groups by coclass. My work involves a great deal of computation in ...
usermath's user avatar
  • 243
26 votes
2 answers
1k views

Is there an algorithm which halts on all inputs that takes as input a finite group ($p$-group if you like) and outputs a finite presentation of the cohomology ring (with trivial coefficients $\mathbb{...
Joshua Grochow's user avatar
  • 3,790
3 votes
0 answers
200 views

Let $S$ and $S'$ be subsets of size $k$ of $\mathfrak{S}_n$. Are there any necessary or sufficient conditions to determine whether or not $S$ and $S'$ yield isomorphic Cayley graphs? Assuming we are ...
Anthony Labarre's user avatar
  • 1,164
9 votes
1 answer
680 views

Let $G$ be a finite group and $\mathcal{L}(G)$ its subgroup lattice. Let $\mu$ be the Möbius function on $\mathcal{L}(G)$. The reduced Euler characteristic of the order complex of the coset poset $\{ ...
Sebastien Palcoux's user avatar
  • 27.6k
1 vote
0 answers
136 views

Let $H_3(\mathbb{Z})$ be the discrete Heisenberg group generated by $x=\begin{pmatrix} 1 & 1 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\\ \end{pmatrix},\ \ y=\begin{pmatrix} 1 & 0 &...
user avatar
9 votes
1 answer
226 views

Let $X$ be a set, and let $F(X)$ be the free group generated by $X$. I will say that an element of $F(X)$ is positive if it is in the monoid generated by all the conjugates in $F(X)$ of every member ...
user49822's user avatar
  • 2,198
2 votes
0 answers
94 views

Let $r\geq 2$. Let $N$ be the normal subgroup of $B_{n}$ generated by conjugates of $\sigma_{i}^{2r}$. Then is the word problem in the quotient group $B_{n}/N$ solvable (in polynomial time)? ...
Joseph Van Name's user avatar
  • 29.1k
3 votes
0 answers
150 views

Tits has proved that a finite simple group $G$ with a BN-pair of rank $n \ge 3$, is of Lie type. Let $B$ be the Borel subgroup and $(W,S)$ the Coxeter system. The subset lattice of the set $S$ is ...
Sebastien Palcoux's user avatar
  • 27.6k
1 vote
1 answer
373 views

Let $G$ be a finite group. Define the Hurwitz action of $B_{n}$ on $G^{n}$ by letting $(x_{1},...,x_{n})\sigma_{i}=(x_{1},...,x_{i}x_{i+1}x_{i}^{-1},x_{i},x_{i+2},...,x_{n})$. I wonder what algorithms ...
Joseph Van Name's user avatar
  • 29.1k
0 votes
0 answers
115 views

Suppose I have a finite group $G.$ How hard is it to find the (a?) minimal degree permutation representation of $G?$ The second part of the question is: is there a table of such (hopefully for ...
Igor Rivin's user avatar
  • 97.7k
8 votes
0 answers
460 views

Let $[H,G]$ be a Boolean interval of finite groups (i.e. the lattice of intermediate subgroups $H \subseteq K \subseteq G$, is Boolean). For any element $K \in [H,G]$, let $K^{\complement}$ be its ...
Sebastien Palcoux's user avatar
  • 27.6k
4 votes
4 answers
565 views

The rank $n$ boolean lattice $B_{n}$ is the subset lattice of $\{1,2, \dots , n\}$. The lattice $B_{3}$ is the following: Question: What are the rank $3$ boolean intervals of the form $[H,G]$, with $...
Sebastien Palcoux's user avatar
  • 27.6k
5 votes
0 answers
114 views

Let $G$ be a finite group and $\mathcal{L}(G)$ its subgroup lattice. Let $s(n):= max\{|\mathcal{L}(G)| \text{ for } |G|=n \}$. There is an OEIS page for the sequence $s(n)$: A018216 1, 2, 2, 5, 2, ...
Sebastien Palcoux's user avatar
  • 27.6k
2 votes
2 answers
419 views

This post is a relative version of General bound for the number of subgroups of a finite group Let $[H,G]$ be a interval of finite groups with $|G:H| = n$. Question: What is a good upper-bound of $|[...
Sebastien Palcoux's user avatar
  • 27.6k
2 votes
0 answers
182 views

The rank $n$ boolean lattice $B_n$, is the subset lattice of $\{1,2, \dotsm n \}$. Let $[H,G]$ be a boolean interval of finite groups. Its Euler totient is defined by $$\varphi(H,G):=\sum_{K \in ...
Sebastien Palcoux's user avatar
  • 27.6k
4 votes
1 answer
456 views

Let $G$ be the reflection group of a regular, 4-dimensional, hyperbolic honeycomb. I would like to find a family $H_i < G$ of finite-index, torsion-free subgroups of $G$, so that I can represent ...
Nikolas Breuckmann's user avatar
  • 83
1 vote
0 answers
101 views

Let $[H,G]$ be a rank $n$ boolean interval of finite groups (i.e. $[H,G] \simeq B_n$ as lattice). Let the set $E = \{ g \in G \ | \ \langle H,g \rangle = G \}$ Remark: If $g \in E$ then $Hg \...
Sebastien Palcoux's user avatar
  • 27.6k
6 votes
1 answer
697 views

Let $G$ be a finite group and $H$ a subgroup such that the interval $[H,G]$ is a boolean lattice. Let $L_1, \dots , L_n$ be the maximal subgroups of $G$ containing $H$. Let the alternative sum ...
Sebastien Palcoux's user avatar
  • 27.6k

1
2