Articles of gap

where to find the tables of irreducible character of the sporadic simple groups and their automorphism groups?

I need tables of complex irreducible characters of the sporadic simple groups and their automorphism groups $Aut(G)$ for some calculation. Also I need information about maximal subgroups of $Aut(G)$ Where Can I find them ? I checked ATLAS but I am not able to find anything useful. I guess I am missing something. For example […]

Finding all the group elements of a certain order of a finite group

Consider a group $G=\langle P, Z, Q \rangle$ generated $P,Z,Q$ where $$Z=\left[ {\begin{array}{ccc} -1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & -1\\ \end{array} } \right] $$ $$ \quad P = \frac{1}{2} \left [ {\begin{array}{ccc} 1 & \tau^{-1} & -\tau\\ \tau^{-1} & \tau & 1\\ \tau & -1 & \tau^{-1}\\ […]

Check element membership in GAP

Does there exist a proper function in order to check a matrix belongs to GL(3,2) in GAP? I have already searched but I didn’t find any function except “IsElementOfFpGroup”. But GL(3,2) is not FpGroup.

Finding ideals of a finite semigroup machinery (GAP)

As you see, I did a finite semigroup and then try to find its possible ideals in a very basic way (with the hand of a friend): > f:=FreeSemigroup(“a”,”b”);; a:=f.1;; b:=f.2;; s:=f/[[a^3,a],[b^2,a^2],[b*a,a*b^3]];; e:=Elements(s);; w1:=List([1..Size(e)],i->Elements(SemigroupIdealByGenerators(s,[e[i]])));; w2:=DuplicateFreeList(w1);; w3 := Combinations(w2);; for i in [1..Size(w3)] do w3[i] := DuplicateFreeList(Flat(w3[i]));; od;; w3 := DuplicateFreeList(w3); > [ [ ], [ […]

Is there a better way to input an $n$-cycle in GAP?

In GAP, we can generate permutations using, for example alpha:=(1,2,3,4,5); But, this method is not usable if we want to input the permutation $$(1,2,\ldots,n)$$ for some variable $n$. It is possible to use alpha:=PermList(List([1..n],i->(i mod n)+1)); for variable $n$, but this feels clumsy. Question: Is there a better way to input an $n$-cycle in GAP?

Substituting values in multivariate polynomial in GAP

given a polynomial $p(x,y)$ from $\mathbb Z[x,y]$. I want to substitute $x=1$ leaving $y$ as it is. The command Value(p,[1,y]) does not work. Can you give me a hint? After that it gets a little more complicated. Let $$p(x,y)=x^6+y^6$$ After substituting $x=1$ we have $p(1,y)=1+y^6$. Given a natural number $j$. I want to find all […]

Question concerning the GAP package qpa (find all ideals with a certain property)

I have the following question concerning the GAP package qpa. Let $k$ be a fixed finite field and let $Q$ be a fixed quiver. Let $kQ$ denote the associated path algebra. Since $k$ is finite, there are only finitely many admissible ideals $I$ of $kQ$ with the property $I^u=0$ for some fixed natural number $u$. […]

How does the function CycleIndex work in GAP? ( undocumented in GAP )

Background: When I divide a hexagon in six triangles the group $D_6$ works on the triangles. The cycle index of the group action would be in this case $$p(x_1,x_2,x_3,x_6)=\frac{1}{12}(x_1^6 + 3x_1^2x_2^2 + 4x_2^3+2x_3^2+2x_6)$$ Question: How does CycleIndex work in GAP? The doc in GAP notes that CycleIndex is undocumented, but the function exists.

How many groups of order $512$ and $1024$ are there with a center of size $2$?

I did not find a sequence in OEIS about the number of groups of a given order with a center of size $2$. For the first few powers of $2$, the numbers are : $2$ : $1$ group $4$ : $0$ groups $8$ : $2$ groups $16$ : $3$ groups $32$ : $10$ groups $64$ […]

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 […]