Articles of gap

About subsets of finite groups with $A^{-1}A=G$ or $AA^{-1}=G$?

Regarding to the problems Does $A^{-1}A=G$ imply that $AA^{-1}=G$? and Is it true that if $|A|>\frac{|G|}{2}$ then $A^{-1}A=AA^{-1}=G$?, we are looking for some subsets $A$ of $G$ with $\lfloor \frac{|G|}{2}\rfloor-2 \leq |A|\leq \lfloor \frac{|G|}{2}\rfloor$ such that $A^{-1}A=G$ and $AA^{-1}\neq G$ or $AA^{-1}=G$ and $A^{-1}A\neq G$. Do such $G$ (non-abelian) and $A$ exist? We propose the […]

Graphical interface in GAP

Is there any graphical interface in GAP? Something like RStudio for R or WxMaxima for Maxima. I’m using GAP under a Linux system. Thanks

Constructing a quotient ring in GAP using structure constants

I need to construct the following ring in GAP: $$Z_4(u) / \langle u^2-2u=0 \rangle. $$ This is what I tried and it didn’t work: gap> R:=PolynomialRing(Integers mod 4,”u”);AssignGeneratorVariables(R); <monoid>[u] #I Global variable `u’ is already defined and will be overwritten #I Assigned the global variables [ u ] gap> I:=Ideal(R,[u^2-2*u]); <two-sided ideal in <monoid>[u], (1 […]

Find generators of a group in GAP

This is a question in the mathematical software called GAP: What is the command for displaying all the generators of a given group? I have been searching around but yet not found anything helpful, so I am hoping I will get a quick response here.

Using GAP to compute the abelianization of a subgroup

Let $K$ be the group generated by four elements $x_1,\cdots,x_4$ with relations that each generator commutes with all its conjugates. (An equivalent relation is, any simple commutator with repeated generator is trivial; for example, $[[x_2,[x_1,x_3]],x_3]=1$.) It can be proved that $K$ is finitely presented. Let $A$ be the subgroup of $K$ generated by the following […]

Is there a computer programm or CAS (maybe GAP?) that can calculate with projective (indecomposable) A-modules (A is a finite dimensional k-algebra)?

I have the following question(s): I have an “Algebra-With-One” $R$ as a subalgebra of a full matrix algebra in GAP. Furthermore, I have 5 primitive orthogonal idempotents $e_1,…,e_5$, which sum up to $1_R$ (the identity matrix). I would like to compute the projective indecomposable modules $P_1=e_1R,…,P_5=e_5R$ with GAP (or another computer programm (e.g. SAGE) which […]

Computing Invariant Subspaces of Matrix Groups

Does anyone have a program written in Mathematica (or SAGE or GAP) that computes the invariant subspace lattice of a matrix group?

How getting the unitarized irreducible representations with GAP?

The function IrreducibleRepresentations on GAP gives non-necessarily unitary representations, for example: gap> R:=IrreducibleRepresentations(SymmetricGroup(5));; gap> m:=(2,3)^R[5];; gap> Display(m); [ [ 1, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 1 ], [ -1, -1, 0, -1, 0 ], [ 0, 0, 1, 0, 0 ] ] A […]

Simplifying Relations in a Group

Let $K$ be the group generated by four elements $x_1,\cdots,x_4$ with relations that any simple commutator with repeated generator is trivial; for example, $[[x_2,[x_1,x_3]],x_3]=1$. As I have asked here, I am trying to use GAP to do some calculation with $K$. Now one problem is that there are too many relations. For example, all following […]

Are the primitive groups linearly primitive?

A transitive permutation group $G \subset S_n$ is primitive if $G_1 \subset G$ is a maximal subgroup. A finite group $G$ is linearly primitive if it has a faithful complex irreducible representation. Question: Are the primitive finite groups linearly primitive? Remark: I’ve checked by a GAP computation that it’s true for $n=[G:G_1] \le 200$ and […]