Character theory of 22-Frobenius groups.

This is a crosspost of my (slightly longer) question on MSE since I’m not getting any responses there. Definition. Let G be a finite group and F1=FitG and F2/F=Fit(G/F1). If F2 is a Frobenius group with kernel F1 and G/F1 is a Frobenius group with kernel F2/F1, we say that G is 2-Frobenius. I have … Read more

The center of a(n endomorphism) ring is a PID

Let A be a torsion-free Abelian group, and let End(A) be its endomorphism ring. Denote by R≐Z(End(A)) the center of End(A). What would be necessary and/or sufficient conditions on A for R to be a PID? Thank you. Edit Following user89334’s comment, indeed, this is true if A is a finitely generated group. The core … Read more

Endomorphism of Brandt Semigroup Bn(G)B_n(G), where GG is a finite group

I want to show that End0(Bn(G))=∪ϕσ,g∪CI(Bn(G)), where ϕσ,g:Bn(G)→Bn(G) is an endomorphism is defined by (i,a,j)ϕσ,g=(iσ,ag,jσ) and σ∈Sn and g∈End(G), CX is the set of all constant map on X and I(Bn(G)) is the set of all idempotents in Bn(G). I have proved that ϕσ,g and constant maps are endomorphism, so ∪ϕσ,g∪CI(Bn(G))⊆End0(Bn(G)) we have proved the … Read more

Name for a regular band

Is there a name for regular bands that satisfy $xyx=yx$ for all $x$,$y$? Answer A right regular band. The dual notion of left regular bands has been used to study random walks on hyperplane arrangements and oriented matroids. The paper of Ken Brown on Semigroups, rings and Markov chains is a good place to start. … Read more

Left- and right-sided principal ideals of quaternions have same index?

One fact about the Lipschitz integers (quaternions of the form $a + bi + cj + dk$ where $a, b, c, d$ are integers) is that the left-sided ideal generated by any element $Q$ has the same index in the additive group as does the right-sided ideal generated by $Q$. I know this is a … Read more

Stanley-Reisner ring of a simplicial complex is a functor?

Let K bea field and [n]={1,…,n} and K[x]=K[x1,…,xn]. For σ={i1,…,ik}⊆[n], denote xσ=xi1⋯xik=∏i∈σxi∈K[x]. Let Δ and Δ′ be (abstract finite) simplicial complexes on [n] and [n′] respectively. Let f:Δ→Δ′ be a simplicial map, i.e. a map f:[n]→[n′] for which ∀σ∈Δ:f(σ)∈Δ′ holds. The Stanley-Reisner ideal of Δ is IΔ:=⟨⟨xσ;[n]⊇σ∉Δ⟩⟩, and Stanley-Reisner ring of Δ is K[Δ]:=K[x]/IΔ=K[x1,…,xn|xσ;[n]⊇σ∉Δ]. The … Read more

Primitive elements in group hopf algebras over fields of non-zero characteristic

An element x of a Hopf algebra H, is called a primitive element if Δ(x)=1⊗x+x⊗1. The set of primitive elements of H is denoted P(H). It can be shown that: “If H is a k-Hopf algebra of finite k-dimension and k is a field of characteristic zero, then P(H)={0}” The proof of the above fact, … Read more

Transitivity of discriminant for flat algebras

Sorry if the question doesn’t feed this site, I’m reposting it from MSE. Nobody answered it there and I couldn’t find the proof in general case(whenever it was mentioned the proof was referred to as a known fact), all I found was the case of Dedekind domains. Let A be an finite flat R-algebra and … Read more

Obstruction for two subgroups to be conjugated by an automorphism

Altough this sounds as a very basic question, I didn’t receive any answer on stack exchange and by people more knowledgeable than me Take p a prime number and P an abelian finite p-group. Let A,A′ be subgroup of P such that A≃A′ and P/A≃P/A′ as groups. Can I conclude that there is ϕ∈Aut(P) such … Read more

Regular functions on a product of varieties

Let $k$ be an algebraically closed field and let $X$, $Y$ be varieties over $k$. Let us denote by $\mathcal{O}(X)$ and $\mathcal{O}(Y)$ the $k$-algebra of regular functions on $X$ and $Y$ respectively. There exists a natural homomorphism of $k$-algebras: $$ \theta \colon \mathcal{O}(X) \otimes_k \mathcal{O}(Y) \to \mathcal{O}(X \times Y) \, , \quad f \otimes g … Read more