All Questions
34 questions
10
votes
2
answers
868
views
When are two semidirect products of two cyclic groups isomorphic
(I have posted this question in Math Stack Exchange, only to have received no answer.)
It is well known that a semidirect product of two cyclic groups $C_m$ and $C_n$ has the form
$$
C_m \rtimes_k C_n ...
- 525
5
votes
2
answers
274
views
Glauberman-Thompson normal $p$-complement theorem for $p=2$
I asked this question on Math StackExchange yesterday. As suggested by Professor Derek Holt, this question may be more suitable for this site. So I ask this question here again, but more details and ...
- 151
2
votes
1
answer
161
views
When are elements of a (perfect) semidirect product simple commutators?
I am migrating this question from math stackexchange...
I have a semidirect product and I have shown that it is perfect. However, I would like to know whether every element is a simple commutator (...
- 23
4
votes
1
answer
273
views
Wedderburn decomposition of special linear groups
$\DeclareMathOperator\SL{SL}\newcommand\card[1]{\lvert#1\rvert}$I want to study about Wedderburn decomposition of group algebra $k\SL(n,\mathbb{F}_p)$ where $k$ is either an algebraically closed field ...
- 249
5
votes
1
answer
448
views
Is any finite-dimensional algebra a sub-algebra of a finite-group algebra?
For $A$ a finite-dimensional algebra over a field $K$
Does there exist a finite group $G$, such that $A$ is a sub-algebra of $K[G]$ ?
Where $K[G]$ denotes the group-algebra of $G$ over $K$.
In case ...
- 188
8
votes
1
answer
198
views
Is there always a simple module whose Green correspondent is a simple module under some conditions?
Let $G$ be a finite group and $KG$ its group algebra over some field $K$ with $\mathrm{char}\ K$ dividing the order of $G$. It's well-known that the Green correspondence is compatible with the Brauer ...
- 775
9
votes
1
answer
507
views
When is the augmentation ideal projective as RG-module?
Let $G$ be a finite group and let $R$ be a commutative ring.
I'd like to ask, if there is a theorem of the following kind:
The augmentation ideal $I_G$ is projective as RG-module, if and only if ... ?...
- 1,755
0
votes
1
answer
426
views
Generators of $SL(n,\mathbb F_2)$? [closed]
Consider the invertible matrices in $\mathbb F_2^{n\times n}$ which are a multiplicative group structure. Is there a finite set of $2k$ (at a $k\in\mathbb Z_{\geq1}$ independent of $n$) generators for ...
- 13.9k
5
votes
0
answers
424
views
A lemma concerning conjugations and normal subgroups (related to a theorem of Frobenius)
In the paper "On A Theorem of Frobenius" written in 1969, Prof. Richard Brauer, for the first time, presented a character-free proof to the Frobenius theorem (i.e. counting the number of ...
- 51
1
vote
1
answer
344
views
Bound for order of a group depending on number of elements of maximal order
This question has been partly answered in MSE, see here.
In a paper On the Number of Elements of maximal order in a Group, it is proven that an arbitrary group $G$ with a finite number of elements of ...
- 181
8
votes
0
answers
251
views
When does a semisimple $\mathbb{C}$-algebra come from a group?
Let $\mathcal{A}$ be a semisimple $\mathbb{C}$-algebra. By the Artin-Wedderburn theorem, it is isomorphic to a direct product of matrix algebras:
$$ \mathcal{A} = \prod_{i=1}^m M_{n_i}(\mathbb{C})$$
...
- 297
6
votes
2
answers
235
views
Is there a CAS that can solve a given system of equations in a finite group algebra $kG$?
Let $k$ be a finite field with char$(k)=p>0$. Let $G$ be a finite group.
Consider the group algebra $kG$.
I would like to solve a given system of equations in $kG$.
Question:
Is there a computer ...
- 1,755
6
votes
0
answers
259
views
Diameter of finite rational matrix groups
Suppose $G$ is a finite subgroup of $\mathrm{GL}(n,\mathbb{Q})$.
For a set $\mathcal{M} \subseteq G$ that generates $G$, define the $\mathcal{M}$-diameter $\mathit{diam}(G, \mathcal{M})$ of $G$ to be ...
- 165
6
votes
1
answer
360
views
Zero divisors in complex group algebras of residually finite groups
Conjecture. There exists a function $f:\mathbb{N} \rightarrow \mathbb{N}$ such that if $\alpha$ and $\beta$ are non-zero elements of the complex group algebra $\mathbb{C}[G]$ of a finite group $G$ ...
- 3,738
4
votes
0
answers
218
views
Conjugacy class representatives for the automorphism group of a finite abelian group
Given a finite abelian group $A$, I'd like a list of conjugacy class representatives for its automorphism group ${\rm Aut}(A)$.
In fact, it's not important that I have exactly one representative from ...
- 286
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
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
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
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
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
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
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
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
4
votes
1
answer
1k
views
Maximal ideal of group ring
Let $R$ be a finite commutative ring with identity and $G$ an finite abelian group. Is there any more conditions (on $R$ or on $G $) under which we can characterize maximal ideals of group ring $RG $, ...
- 41
1
vote
2
answers
432
views
Subgroup of Projective general linear group on complete discrete valuation ring
Let $R$ be a complete dvr and $k$ its residue field of positive characteristic.
Let $H$ be a finite subgroup of $PGL_2(k)$ such that the order of $H$ is prime with $char(k)$.
Is there some ...
- 205
1
vote
1
answer
225
views
Is a prime index inclusion of finite groups, separating?
Let $(H \subset G)$ be an inclusion of finite groups.
Let $\{ g_i \ \vert \ i \in I=[1, \dots ,n] \}$ a subset of $G$ of double coset representatives, i.e. $$G = \coprod_{i \in I} Hg_iH$$
On the ...
0
votes
0
answers
143
views
Soluble group algebras and centralizers
Let $K$ be field with $\mathrm{char}(K) = p > 0$ and $G$ a finite group such that $KG$ is soluble. Then the $p$-Sylow-subgroup of $G$ is normal and contains the derived group of $G$ and every $p'$-...
- 581
4
votes
1
answer
686
views
Character theory of $2$-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 $F_1=\text{Fit}\,G$ and $F_2/F=\text{Fit}\left(G/...
- 1,173
1
vote
2
answers
1k
views
Maximal subgroups of a finite p-group
I want to prove the following:
Let $G$ be a finite abelian $p$-group that is not cyclic.
Let $L \ne {1}$ be a subgroup of $G$ and $U$ be a maximal subgroup of L then there exists a maximal subgroup $...
- 21
0
votes
0
answers
133
views
On X-s-permutable subgroups of a finite group
I want to prove Lemma 2.1(1) in the paper On X-s-Permutable Subgroups of a Finite Group by Min Bang SU, Yang Ming LI. It is on the web.
This is my proof.
.
Since $H$ is $X−s−$permutable in $G$, then ...
3
votes
1
answer
315
views
Group ring and left zero divisor II
Let $K$ be a finite field and $G$ be a discrete group.
Is it true that for every $a=e+a_1+\ldots+a_n,b=e+b_1+\ldots+b_m\in K[G]$ with $b_i\neq e,a_j\neq e$ the condition $ab=0$ implies $ba=0$?
...
- 4,705