13
votes
Accepted
Distinct characters with the same character values, outer automorphisms and Galois conjugation
Take $G = S_3 \times S_4$ and consider the unique two-dimensional irreducible representation of $S_3$ and the unique two-dimensional irreducible representation of $S_4$. These have the same character ...
- 148k
12
votes
Accepted
recognition of symmetric groups in GAP
The method (I assume) uses Jordan's theorem, which says that an primitive subgroup of $S_n$ with a cycle of prime order (at most $n-2,$ if memory serves) is either $A_n$ or $S_n.$ You rule out $A_n$ ...
- 96.4k
12
votes
Is It possible to determine whether the given finitely presented group is residually finite with MAGMA or GAP?
It is a consequence of the Adian-Rabin theorem that there is no algorithm that decides, given a finite presentation of a group, whether the group is residually finite.
In a similar spirit, it is ...
- 3,736
11
votes
Accepted
Database subgroups of free group
I think the answer is no. There exists a Magma command $\mathtt{LowIndexNormalSubgroups}$ that does what you want, and it does indeed find generators for each of the subgroups. I believe that a ...
- 37.4k
8
votes
Is there a maximal subgroup of depth 3?
This isn't a full answer.
First let’s translate this into purely group theoretic language.
The vertex at depth $0$ is the trivial $G$-rep, the vertex at depth $1$ is the trivial $H$-rep, the ...
- 28.1k
8
votes
Is It possible to determine whether the given finitely presented group is residually finite with MAGMA or GAP?
To precise Giles Gardam's answer, let me add the following.
The Adian-Rabin theorem shows that residual finiteness is undecidable, by showing that no algorithm can stop exactly on non-residually ...
- 543
7
votes
Lower bound for the order of a simple group with a given class number
Apart from finitely many examples, such results would follow from the work of Liebeck and Shalev (Proc. London Math. Soc. 2005). In Corollary 5.2 of that paper they show that for some constant $c &...
- 43.7k
7
votes
The associated graded algebra of a finite dimensional algebra
It's hard to find references in this generality. But it is worth mentioning the following.
(1) Many properties are lost in forming the associated graded algebra of the radical filtration. For example, ...
- 16.2k
6
votes
Accepted
Get the commands history from GAP system
There are three commands that will do all variations of this:
LogTo("filename.txt") will save all subsequent input and output to a file with the ...
6
votes
Membership to double cosets in free groups
Benois gave a Stallings type algorithm before Stallings to compute membership in any rational subset of a free group on a set $X$. A rational subset $R$ is given by a finite automoaton over the ...
- 38.6k
6
votes
Membership to double cosets in free groups
Concerning implementations in GAP and Magma, there is a GAP package for handling subgroups of free groups, but I don't know whether that has the built-in functionality to do what you want.
I know more ...
- 37.4k
6
votes
Accepted
Membership to double cosets in free groups
Here is a second answer that is just rephrasing @DerekHolt’s answer based on the comments. So upvote his answer first! Let $X$ be a finite alphabet. An inverse automaton is a finite directed graph ...
- 38.6k
6
votes
Accepted
A question about the possibilities of GAP
It is contained. Here is the way I did the calculation with GAP. It ought to work nicer, but there is a stupid technical issue in the way that makes it hard to implement a membership test in a ...
- 1,210
6
votes
Accepted
Computing homology groups with GAP
Graham Ellis would be able to better comment on the correctness of his code for $SL(5,\mathbb Z)$, as he appears to be the author of the HAP package in GAP.
But his code executes quickly and claims to ...
- 44.4k
5
votes
Lower bound for the order of a simple group with a given class number
It has been proved by J. Fulman and R. Guralnick that is $G$ is ``almost simple", (and $F(G) = 1$), then $k(G) < |G|^{0.41}.$ This yields $|G| > k(G)^{2.436}.$ This result needs the ...
- 44.4k
5
votes
Maximal numbers of summands in middle terms of short exact sequences
For question 1:
For a self-injective algebra, if $0\to X\to Y\to Z\to0$ is a short exact sequence with $Y$ and $Z$ indecomposable, then there is a short exact sequence $0\to\Omega Z\to X\oplus P\to Y\...
- 35.2k
5
votes
Accepted
Cayley graphs on $Z_{11}$ and $Z_p$
If you are interested in graphs (not digraphs), then the elements of the connection set must come in pairs, so you are only looking at subsets
$$
C \subseteq \{\pm1, \pm2, \pm3, \pm4, \pm5\}.
$$
...
- 12.7k
5
votes
On the sum of the subgroup orders of a finite group
Partial answer:
The answer to Q1 is "yes" if one restricts $G$ to supersolvable groups or more generally groups that satisfy the converse of Lagrange's theorem.
If $G$ has the property ...
- 9,650
5
votes
Accepted
GAP versus SageMath for branching to Lie subgroups
I don't know GAP but Sage has a nice tutorial for branching and is quite usable. It is however, slower than LiE which is on the other hand quite "basic" i.e. it requires you to write the branching ...
- 8,597
5
votes
Accepted
Is there a maximal subgroup of depth 3?
As Noah points out, you are looking for some (core-free) maximal subgroup $H<G$ such that $1_H^G$ has nonzero inner product with every irreducible character.
Say $G=L_2(p)$ with $p$ prime and $p \...
- 3,571
5
votes
Irreducible tensor product representations in finite simple groups
This is not an answer, but it reminded me of a beautiful general argument of R. Brauer (which I first saw in a paper of David Wales, (credited to Brauer, though I am not sure whether Brauer ever ...
- 44.4k
4
votes
Finding all submodules
The generic method for submodules would be to write down a matrix representation and to use MeatAxe tools -- in GAP there is e.g. a function MTX.BasesSubmodules. ...
- 1,210
4
votes
Algorithm for finding quiver algebras
Here are some observations:
$A$ is just a tensor algebra $T(V)$ for $V=K^r=span\{x_1,\ldots,x_r\}=J/J^2$. Let's work with that because choosing bases is evil.
Every morphism $\phi: A/I \to A/I'$ ...
- 9,650
4
votes
Obtaining quiver and relations for finite p-groups
Here is a computation of the quiver. I don't know how to get the relations. Let $G$ be a finite $p$-group and $K$ the $p$-element field. Note that the trivial module is the unique simple module and ...
- 38.6k
4
votes
Accepted
Testing whether a module generates $K_0(\mbox{mod-}A)$
Suppose that $M = \oplus_{i=1}^n M_i$ with $M_i$ being indecomposable and assume that $M_i \not\simeq M_j$ for $i\neq j$ (that is, $M$ is basic) over a finite dimensional algebra $A$. Define a matrix $...
- 1,244
4
votes
On the sum of the subgroup orders of a finite group
Here is a small remark: let $n$ be a number which is not abundant (that is, the sum of the proper divisors of $n$ is at most $n$). Suppose further that there is a non-cyclic finite group $G$ of order $...
- 44.4k
4
votes
Irreducible tensor product representations in finite simple groups
Also not an answer, just a hint on how the GAP code can be made a bit faster, for future reference: GAP actually "knows" the character tables for the alternating groups. For some reason I ...
- 5,654
3
votes
Accepted
Preprojective algebra of finite dimensional algebras
For Question 2: In QPA one can do the following using the latest uploaded extensions of QPA:
...
- 1,244
3
votes
Accepted
Problem while multiplying under a set of relators
Algorithms for finitely presented groups are hard -- generically problems, such as testing whether a word represents the identity (or finding a shortest word expression) do not have (they cannot exist ...
- 1,210
3
votes
Accepted
A global code for the character table of PSL(2,q)
The quickest way to display the character table for a fixed prime-power $q$ is:
Display(CharacterTable( "PSL", 2, q)
The generic character table is in ...
Only top scored, non community-wiki answers of a minimum length are eligible