# Questions tagged [computational-group-theory]

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122 views

### Algorithm to compute automorphism group of a finite group

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 ...
113 views

### Tools for computing from group presentations

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 ...
29 views

### Rearrangements of Jordan–Holder quotients

Suppose we have a composition series $$0=M_n\subset M_{n-1}\subset\cdots\subset M_1\subset M_0$$ of $R$-modules, or one of groups: $$1=G_n\lhd G_{n-1}\lhd\cdots\lhd G_1\lhd G_0$$ (not necessarily ...
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### Algorithm for root system of Coxeter group generated by permutations

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 ...
193 views

### A global code for the character table of PSL(2,q)

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): ...
199 views

### Isomorphism of semidirect products of surface groups

Recall that the fundamental group of a closed Riemann surface of genus $h$ has the presentation $$\Pi_h= \langle a_1, \,b_1, \ldots, a_h,\, b_h \; | \; [a_1, \, b_1]\ldots [a_h, \, b_h]=1 \rangle.$$ ...
291 views

### Are the character degrees determined by the conjugacy class sizes?

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)...
194 views

### Can MAGMA compute almost projective $kG$-homomorphisms?

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....
179 views

### Where or how can I find matrix representatives of the conjugacy classes of Conway's group Co₀?

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. ...
682 views

### Groups without factorization

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 (...
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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 ...
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### Generating sets of the symmetric group that yield isomorphic Cayley graphs

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 ...
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### Detecting/Characterising positive elements in free groups

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 ...
63 views

### Is the word problem in the braid group quotient $B_{n}/N$ solvable where $N$ is the normal subgroup generated by conjugates of $\sigma_{i}^{2r}$?

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)? ...
123 views

### Extension of Tits' theorem on groups with a BN-pair of rank ≥ 3

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 ...
248 views

### How quickly can one compute the Hurwitz action of braid groups on finite groups?

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 ...
100 views

### minimal permutation representations [duplicate]

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 ...
419 views

### A relation between intersection and product on Boolean interval of finite groups

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 ...
356 views

### What are the rank 3 boolean intervals [H,G], with G simple group?

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 $[H,G]$, ...
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, ...