gr.group theory – Is there overwhelmingly more finite monoids than finite spaces?

A function $f:mathbb{Z}_{geq 1}tomathbb{Z}_{geq 1}$ overwhelms $g:mathbb{Z}_{geq 1}tomathbb{Z}_{geq 1}$ if for any $kin mathbb{Z}_{geq 1}$ the inequality $f(n)leq g(n+k)$ holds only for finitely many $ninmathbb{Z}_{geq 1}$.

Does the number of non-isomorphic monoids of cardinality $n$ overwhelm the number of non-homeomorphic topological spaces of cardinality $n$?

gr.group theory – Computable change in minimum word length of subgroup elements

Let $G$ be an infinite finitely generated group. Fix a finite generating set for $G$.

Define $mathrm{len}_G:Gtomathbb{Z}_{geq 0}$ by sending $g$ to the minimum length of a word in the generators and their inverses equal to $g$.

Let $Hsubset G$ is an infinite finitely generated subgroup. Fix a finite generating set for $H$.

Question. Under what conditions is there a computable function $m colon mathbb{Z}_{geq 0}tomathbb{Z}_{geq 0}$ such that for all $hin
H$
the inequality $$ mathrm{len}_H(h)leq m(mathrm{len}_G(h)) $$
holds?

gr.group theory – How to classify rings by combinatorial structures?

There are many ways to encode information about algebraic structures such as groups, rings, etc… in combinatorial form. For example the Cayley graph of a group with a subset of generators, or the various graphs associated to rings, as can be found in, e.g., the answers to
Why do we associate a graph to a ring?. So I was wondering about the converse questions, which for groups and rings take the form:

First question(s): Given $X$ a graph is there a way to discover, intrinsically, constructively and algorithmically, whether it is the Cayley graph of a group? How to recover the group structure from the graph? Is there a unique group $G$ such that $X = CG(G)$, the Cayley graph of $G$? How to find such a group $G$? Which graphs are the Cayley Graphs of some Group?

Second Question: Is there a combinatorial structure (such as a system of graphs) associated to rings (or algebra or module of an algebra) from which you can recover the full ring (or module) in a similar manner as in the first question? Preferably in an intrinsic, constructive and algorithmic way. Assuming one could find such a combinatorial category, how to find out which objects in it are the objects associated to rings (or modules)?

I would also be interested in considering similar types of questions for general well-known algebraic structures (some kind of combinatorial informational encoding for these algebraic structures) in the sense that you can define precisely combinatorial structures out of algebraic structures, intrinsically constructively and algorithmically, but from which you can recover the original structure, also intrinsically constructively and algorithmically and intrinsically.

For groups, there is positive answer given by Sabidussi’s theorem, as mentioned in https://en.wikipedia.org/wiki/Cayley_graph#Characterization, which characterizes graphs which are Cayley Graphs of groups. This theorem would suffice in terms of instrisic, constructive and algorithmic profile of the proof, for question 1.

I would be satisfied with partial answers.

gr.group theory – Rack cohomology as derived functor cohomology

Let $X$ be a rack and $A$ be an $X$-module. By this paper, p. 33, we can associate a cochain complex $C^bullet(X,A)$ to the pair $(X,A)$. This complex is explicitly defined by a differential $d$. I wonder if the cohomology $H^bullet(X,A)$ of the complex has an interpretation as derived functor cohomology. What functor from $X$-modules to $X$-modules do we have to derive? And how to show then the equivalence of the two definitions? I think the analogy to group cohomology is not very helpful, or can we somehow define the invariants of an $X$-module and make it fit?

gr.group theory – Group cohomology with coefficients in a graded module

I am working in a problem group cohomology and nailed it down to compute a cohomology using an spectral sequence argument. The situation is as follows:

Let $G = C_4 = langle sigma rangle$ be the cyclic group of order $4$, $k = mathbb{F}_2$ and $M^*=M^0 oplus M^1 oplus M^2$ be a graded module where $M^1 = M^2 = k$ is the trivial $G$-module and $M^1 = k oplus k$ be the $G$-module where $sigma(a,b) = (b,a)$.

To compute $H^*(G, M^*)$, we can use a spectral sequence argument with $E_2^{p,q} =H^p(G,M^q)$. Since $M^0$ and $M^2$ are trivial, $E_2^{p,q} = H^p(G,k) otimes M^q$ for $q=0,2$.

When $q = 1$, we can use Shapiro’s lemma to show that $E_2^{p,1} = H^p(C_2;k) cong k(t)$ where $C_2 = langle sigma^2 rangle$.

However, I have no leads on how to compute the differential $d_2:E_2^{p,1} = H^p(C_2,k) rightarrow E_2^{p+2,0} = H^{p+2}(C_4,k)$.

I appreciate any input in this computation, or if there is any alternative to describe $H^*(G,M^*)$.

gr.group theory – Is every permutation group on $n$ letters the symmetry group of a set of $n$ points in some euclidean space?

No. If $G$ is $2$-transitive $-$ or even transitive on pairs $-$ then
$X$ must be the set of vertices of a regular simplex,
which has isometry group $S_n$. But there are plenty of
examples of $2$-transitive groups $G$ properly contained in $S_n$
(such as the $ax+b$ group if $n$ is a prime power and $n>3$).