All Questions
Tagged with ra.rings-and-algebras gr.group-theory
309 questions
10
votes
1
answer
274
views
A flatness result of Fiedorwicz for amalgamated free products of monoids in connection with classifying spaces of monoids
In Lemma 5.2(a) of Z. Fiedorowicz, Classifying Spaces of Topological Monoids and Categories American Journal of Mathematics Vol. 106, No. 2 (Apr., 1984), pp. 301-350 the author proves the following.
...
- 38.6k
7
votes
2
answers
485
views
Using Dunwoody's results on cohomological dimension to learn about a von Neumann regular group ring
Just recently I've stumbled across Warren Dicks' book Groups, trees and projective modules (1980) and I was pretty stunned. I know nothing of group cohomology, but I gather the "tree" ...
- 1,507
2
votes
0
answers
127
views
Multiplicative subgroups of $GL(V)$ which are almost additively closed
Edit:
According to comments of YCor and Vincent, I revise the question.I appreciate their comments:
Let $G$ be a group.
We say that $G$ is a special group if it is isomorphic to a subgroup $H$ ...
- 356
10
votes
1
answer
1k
views
Equivalent descriptions of Coherent Groups
Attending a series of lectures, I have recently been exposed to the notion of Coherent groups, defined as following:
Def: A group $G$ is called Coherent if every finitely generated subgroup $H$ of $G$...
- 493
2
votes
1
answer
255
views
Does anyone have a copy of Salce's paper "Cotorsion theories for abelian groups"?
The paper "Cotorsion theories for abelian groups" by L. Salce, was published in 1979 in Symposia Math. 21, pages 1-21. According to Google Scholar, it's been cited 233 times, and I keep seeing ...
- 30.3k
5
votes
1
answer
548
views
Question about denoting/designating of algebraic structures
I saw this image on Wikipedia (Template:Group-like structures, current revision):
Since there are five "properties" that we can have (in this context), namely: totality, associativity, identity, ...
user114642
-1
votes
1
answer
218
views
A Rng of rotations?
It's very standard to view rotations about the origin in $\mathbb{R}^2$ as a group $\mathbb{SO}(2)$, with the zero rotation as an identity and composition of rotations as addition. This can also be ...
- 10.1k
1
vote
0
answers
155
views
Deduce Schur-Zassenhaus theorem from Wedderburn-Malcev theorem
Is it possible to derive Schur-Zassenhaus theorem from Wedderburn-Malcev theorem? For the existance part I have the following idea: Let $G$ be a finite Group and $N$ an abelian Hall $p$-subgroup. Let $...
- 581
3
votes
0
answers
65
views
Intersections of generating sets of subalgebras
Let $A$ be a finitely generated, finitely presented, Noetherian, unital algebra over the complex numbers, which has no zero divisors. We do not assume that $A$ is commutative however.
Moreover, let $...
8
votes
0
answers
685
views
What classifies involutive automorphisms on finite groups? What classifies involutions on finite based rings?
Groups
Let $G$ be a finite group. An involutive automorphism on $G$ is an automorphism $i\colon G \to G$ such that $i^2 = 1_G$.
Question 1. What classifies involutive automorphisms on a given (non-...
- 5,565
8
votes
2
answers
255
views
Finding a compatible multiplication for a given group
If you are given an abelian group $\ (G, +)$, is there some algorithm to find all possible semigroups $\ (G, ×)$, such that $\ (G, +, ×)$ is a ring?
If not, can you at least decide, if the ring must ...
- 3,738
36
votes
1
answer
3k
views
Whence “homomorphism” and “homomorphic”?
Today homomorphism (resp. isomorphism) means what Jordan (1870) had called isomorphism (resp. holoedric isomorphism). How did the switch happen?
“Homomorphic” (and “homomorphism” as “property of being ...
- 31.5k
4
votes
1
answer
434
views
Regarding a new algebraic structure
By "left semigroup-joined-semigroup" I mean an algebraic structures $(S,\cdot,*)$ such that both $\cdot,*$ are associative, and the following property holds (see this )
$$
x*(y\cdot z)=x*y*z\;\; ; \;...
- 995
4
votes
1
answer
154
views
Right reversibility of submonoids of nilpotent groups
Let $G$ be a finitely generated group (optionally torsion-free). Let $N$ be a submonoid of $G$ (that is, a subsemigroup with $1$).
A (cancellative) monoid/semigroup $S$ is right reversible if for ...
- 4,679
3
votes
1
answer
137
views
Subalgebras with finite codimension
In group theory it is well-known that every subgroup of finite index contains a normal subgroup of finite index. It is not true in general that for Lie algebras every subalgebra of finite codimenslon ...
- 379
6
votes
2
answers
483
views
In the group ring $\mathbb{Z}_p [G]$, what elements satisfy $(\sum a_g g)^p = \sum a_g g^p$?
Here $\mathbb{Z}_p$ is the ring of integers in $\mathbb{Q}_p$.
Preferably I would want to know this for a general group $G$, but I have been concentrating on the case $G = (\mathbb{Z} / p^n \mathbb{Z}...
- 153
9
votes
1
answer
893
views
Kaplansky conjecture (consequences)
The Kaplansky conjecture says that: for any field $F$ and any torsion free group $G$, the group ring $F[G]$ does not have nontrivial idempotent elements.
Questions
Do we assume that $F$ has any ...
- 641
12
votes
1
answer
2k
views
Relationship between the Witt algebra and vector fields on the circle
I have seen some (apparently contradictory) claims by mathematicians and physicists regarding the existence of an (infinite dimensional) Lie group whose Lie algebra is the Virasoro algebra.
The ...
- 1,329
16
votes
3
answers
3k
views
Are there other semidirect product/crossed products in other areas
Suppose $(O, G, \alpha)$ is a triple where $O$ is some mathematical object, $G$ is a group and $\alpha : G \rightarrow Aut(O)$. Many different areas of mathematics study such triples. However, I only ...
- 3,984
0
votes
1
answer
75
views
How do we prove that the following implication in semiring? [closed]
Let $G$ be a group. Clearly the power set $(\mathcal{P}(G),\cup,. )$ is the semiring, where $\cup$ means ordinary union and '.' is defined as $$AB= \left\lbrace ab \in G \mid a\in A\mbox{ and } b\in B ...
2
votes
1
answer
151
views
Right socle of a group ring
Let $p$ be a prime number and $n$ a positive integer. I want to know what is the (right) socle of the group ring $A=\mathbb Z_{(p)}C_n$, where $\mathbb Z_{(p)}$ is the localization of integers at the ...
- 355
6
votes
2
answers
210
views
groupring morphisms and bialgebra
Let $G_{1}$ and $G_{2}$ be two groups. Suppose that we have a morphism $\mathbb{Z}[G_{1}]\rightarrow \mathbb{Z}[G_{2}] $ of bialgebras is it true that this morphism comes from a morphism of groups $G_{...
- 1,613
10
votes
2
answers
474
views
Groups determined by their group ring and direct products
In the paper [W. Kimmerle - R. Lyons - R. Sandling - D.N. Teague: Composition factors from the group ring and Artin's theorem on orders of simple groups, Proc. London Math. Soc. (3) 60 (1990), no. 1, ...
- 601
24
votes
3
answers
1k
views
Is $\widehat{\mathbb{Z}}[[t]]\cong\widehat{\mathbb{Z}}[[\widehat{\mathbb{Z}}]]$?
Let $\widehat{\mathbb{Z}}[[\widehat{\mathbb{Z}}]] := \varprojlim_{n,m}(\mathbb{Z}/n)[x]/(x^m-1)$ be the complete group algebra of the profinite free group of rank 1. In Corollary 5.9.2 of Ribes-...
- 7,527
15
votes
1
answer
566
views
Torsion-free abelian group $A$ such that $A \not \simeq A \oplus \Bbb Z \simeq A \oplus \Bbb Z^2$
Is there a torsion-free abelian group $A$ such that $A \not \simeq A \oplus \Bbb Z \simeq A \oplus \Bbb Z \oplus \Bbb Z$ (as groups)?
Notice that $\Bbb Z$ is not cancellable, so
$A \oplus \Bbb Z \...
- 1,742
13
votes
2
answers
515
views
Free groups and free restricted Lie algebras
If $G$ is any group and $\gamma_k(G)$ denotes the $k$th term in the lower central series of $G$, then the commutator bracket on $G$ endows
$$\mathcal{L}(G) = \bigoplus_{k=1}^{\infty} \gamma_k(G) / \...
- 44.8k
17
votes
1
answer
419
views
Freeness of tensor product
Let $G$ be a finite group. Is $\mathbb{Z}G\otimes_{Z(\mathbb{Z}G)}\mathbb{Z}G$ free as a $\mathbb{Z}$-module, where $Z$ denotes the centre?
- 355
10
votes
1
answer
569
views
Non-isomorphic Heisenberg groups over rings
Suppose $R_1,R_2$ are finite unital commutative rings. Consider Heisenberg groups $H_3(R_1)$ and $H_3(R_2)$ (upper unitriangular marticies $3 \times 3$).
Proposition. If $R_1 \not\cong R_2$ (as ...
- 411
0
votes
1
answer
121
views
Do we have a one to one correspondence between positive roots and reflections in a Coxeter group?
By the answer of the question, the set of reflections of a Coxeter system $(W,S)$ is given by $R = \{ wsw^{-1} : w \in W, s \in S\}$.
Do we have a one to one correspondence between positive roots and ...
- 6,201
1
vote
1
answer
474
views
Is there any algorithm to find the minimal generating set for $A_{n}$
I know that finite simple groups can be generated by two elements.(See this question on MO)
So as a specific example, Take Alternating group $A_{n}$, $n>4$.
We also know that $A_{n}$ is $(2,3)$ ...
- 591
0
votes
1
answer
123
views
Orthogonal idempotents with sum equal to 1 in $k[G]$ span sub-Hopf algebra
Let $G$ be a finite group. Let $B$ be a set of orthogonal non-zero idempotents with $|B| \leq |G|$, s.t. $\sum_{b \in B}b =1_{kG}$. Is it known if $B$ spans a sub-Hopf algebra $kH \subseteq kG$?
- 1,813
1
vote
2
answers
613
views
Jacobson radical of group algebra
For a finite group G and a finite field $\mathbb{F}_p$ of characteristic $p$,
J($\mathbb{F}_{p^k} \otimes_{\mathbb{F}_p }\mathbb{F}_p G ) = J(\mathbb{F}_{p^k}G)$? where $J(\mathbb{F}_{p^k}G)$ is the ...
- 11
2
votes
3
answers
545
views
Irreducible representations of $\text{SL}(2, \mathbb{F}_q)$ which don't exist in decomposition?
Let $\mathbb{F}$ be a finite field with $q$ elements, and let $G = \text{SL}_2(\mathbb{F})$. The group $G$ acts linearly on the $2$-dimensional vector space $\mathbb{F}^2$ and fixes the origin $0$. ...
- 29
4
votes
0
answers
117
views
Generators of the symmetric square of the group ring of an abelian group
Let $A$ be an abelian group and $R=\mathbb{Z}[A]$- its group ring. Denote by $I$ an ideal of $R$ given by a kernel of the map $R\longrightarrow \mathbb{Z} \oplus A,$ sending $[r]$ to $(1,r)$. Next, ...
- 4,316
2
votes
0
answers
1k
views
Lifting of group homomorphisms
I asked this question a few days ago on math stackexchange but didn't get any answer so I thought I post it here too (see here):
In my first course on algebraic topology I heard about the following:
...
- 721
3
votes
4
answers
610
views
Factorization in the group algebra of symmetric groups
Let $S_n$ be the symmetric group on $\{1, \ldots, n\}$. Let
\begin{align}
T=\sum_{g\in S_n} g.
\end{align}
Are there some references about the factorization of $T$?
In the case of $n=3$, we have
\...
- 6,201
5
votes
2
answers
364
views
Have semigroups with actions on themselves that have a dual to the compatibility axiom ever been studied?
For a semigroup $G$ with a left action on itself, the axiom for compatibility becomes:
$$
\forall f,g,h\in G:hg(f)=h(g(f))
$$
Now suppose there is additional axiom, or constraint if you prefer, ...
- 505
7
votes
0
answers
372
views
"Factorisation" in special linear groups over rings of integers
It is known that for any number field $F$ with infinitely many units (i.e. $F$ is not $\mathbb Q$ or an imaginary quadratic field) with ring of integers $O$ the special linear group $\mathrm{SL}_2(O)$ ...
- 3,399
2
votes
3
answers
318
views
Realization of irreducible $\mathfrak{S}_d$-modules and the representation theory of Lie algebra
Let $n$ be a positive integer. It is well-known that a method to realize irreducible $\mathfrak{S}_d$-modules is to construct the so-called Specht modules $S^{\mu}$ which are submodules in the so-...
- 159
4
votes
0
answers
167
views
For $P$ $\mathbb{Z}G$-projective, $\mathbb{Q}\otimes P$ is $\mathbb{Q}G$-free
I'm looking for a proof of a theorem of Swan [1, Theorem 3]:
If $G$ is a finite group and $P$ a finitely generated projective $\mathbb{Z}G$-module, then $\mathbb{Q}\otimes_\mathbb{Z}P$ is a free $\...
- 2,019
1
vote
1
answer
271
views
Rank of a locally free $\mathbb Z[G]$-module
This is a basic question. Let $G$ be a finite group, $M$ a finitely generated $\mathbb Z[G]$-module so that the $\mathbb Z_p[G]$-module $M_p$ is free for all prime numbers $p$, i.e. is locally free.
...
- 255
8
votes
0
answers
307
views
How bad can $SK_1$ of a commutative ring be?
For a commutative ring $R$ define $\mathrm{SK}_1(n, R)=\mathrm{SL}(n, R)/\mathrm{E}(n, R)$, the quotient of the special linear group by its subgroup generated by the elementary matrices. When $n\...
- 3,186
11
votes
0
answers
400
views
Detecting a module for the free group algebra on a finite quotient
Let $F_2$ be the free group on two generators $x,y$ and let $R$ be the group algebra $\mathbf{Q}[F_2]$. Let $a,b,c$ be integers. Then define a right $R$-module
$M = R / (ax + by + c) R$.
I am ...
- 19.2k
2
votes
2
answers
496
views
For a ring R, does $GL_n(R)$ embed into $GL_m(F)$ for some field F?
Suppose $R$ is a noncommutative ring. What is the sufficient and necessary condition for $GL_n(R)$ embedding into $GL_m(F)$ for some field $F$?
In particular, what if $R$ is the group ring $\mathbb{Z}...
- 57
2
votes
0
answers
468
views
Semidirect product of semidirect products
For algebraic objects, say for groups $ N,K$ and algebra $A$, if we have the semidirect product of a semidirect product,
$(A \rtimes_{\gamma} N ) \rtimes_{\theta} K$, are there conditions that would ...
- 21
2
votes
1
answer
520
views
A group algebra isomorphism problem
What are the groups $G$ and fields $\Bbb K$ for which $\Bbb K[G]\cong\Bbb K^{|G|}$ holds?
For example $\Bbb R[\Bbb F_2^n]\cong\Bbb R^{2^n}$ holds.
user76479
6
votes
0
answers
219
views
Is always the ratio (number of commuting pairs of elements in a ring or a Lie algebra)/(the size of the ring or the Lie algebra) integer?
(1) Is there a finite nilpotent ring $R$ such that the ratio
$$c(R)=\frac{|\{(x,y)\in R\times R \; | \; xy=yx\}}{|R|}$$
is not integer?
Edit 1: The nilpotent condition is put later.
Edit/Answer: ...
- 3,738
1
vote
0
answers
206
views
The normalizer problem for group rings
I recently studied about The Normalizer problem (NP) which states that given an integral group ring $\Bbb{Z}G$, $N_{\cal{U}}(G)=G\frak{z}$ where $\frak{z}$ denotes centre of $\cal{U}$ = $\cal{U}$$(\...
0
votes
0
answers
308
views
Basic question about power series and complete group algebras
This is a pretty basic question, but I suspect it might be too exotic for math.stackexchange.
Let $\mathbb{Z}_p$ be the $p$-adic integers. For free pro-$p$ group $F_r$ of rank $r$, we can consider ...
- 10.7k