All Questions
Tagged with gr.group-theory finite-groups
1,583
questions
12
votes
2
answers
578
views
The mysterious significance of local subgroups in finite group theory
EDIT 21/12: Even if there are no conclusive answers to these questions, I would very much like to know if anyone has noted and attempted to explain the mysterious significance of local subgroups: are ...
- 917
2
votes
1
answer
210
views
Finitely generated G, such that x^3 = 1 for all x, is finite? [closed]
x^3 = e for any element x in finitely-generated group G. How to prove that G is finite?
- 31
17
votes
1
answer
1k
views
Explicit character tables of non-existent finite simple groups
In connection with the historical development of the classification of finite simple groups, I am interested in a particular aspect that seems to be less well-documented than the main narrative of ...
- 25.5k
6
votes
2
answers
165
views
Finite groups with only one $p$-block
If $G$ is a finite group with a prime $p \big| |G|$, and $G$ has exactly one $p$-block, namely the principal block, can anything be said about the structure of $G$? I am aware that when $G$ has ...
- 163
8
votes
0
answers
366
views
Is this set, defined in terms of an irreducible representation, closed under inverses?
$\DeclareMathOperator\GL{GL}$Let $ H $ be an irreducible finite subgroup of $ \GL(n,\mathbb{C}) $. Define $ N^r(H) $ inductively by
$$
N^{r+1}(H)=\{ g \in \GL(n,\mathbb{C}): g H g^{-1} \subset N^r(H) \...
- 3,429
2
votes
0
answers
120
views
Does every faithful action on a scheme act freely on a dense open subset?
Disclaimer: I have asked this question on math exchange a week ago (here), but sadly to no avail. So I decided to escalate my question:
Let $G$ be a finite group acting faithfully on a smooth quasi-...
- 41
1
vote
1
answer
93
views
Maximal abelian subgroups of an extraspecial group of order $2^{2m+1}$
I've found a proof of the structure of maximal abelian normal subgroups of an extraspecial group of order $2^{2m+1}$ in the book "Endlichen Gruppen I" by B. Huppert but there is a part of ...
1
vote
1
answer
155
views
Suggestions about the set of all irreducible complex character degrees of a finite group
Let $G$ be a finite group, $\operatorname{cd}(G)$ be the set of all irreducible complex character degrees of $G$, and $\rho(G)$ be the set of all prime divisors of integers in $\operatorname{cd}(G)$. ...
- 577
1
vote
0
answers
109
views
$p'$-automorphisms of pro-$p$ groups
Let $p$ be a prime and $G$ be a finitely generated pro-$p$ group admitting a continuous automorphism $\phi$ of finite order relatively prime to $p$. Let $\Phi(G)$ denote the Frattini subgroup of $G$. ...
- 499
5
votes
2
answers
371
views
Cohomology of the adjoint representation of $\mathrm{SL}_2(k)$
Let $k$ be a finite field. Do we always have $H^1(\operatorname{PSL}_2(k), k^3) = 0$, where $\operatorname{PSL}_2(k)$ acts on $k^3$ via the adjoint representation (= conjugation action on trace zero ...
- 35.8k
6
votes
2
answers
224
views
Group homology for a metacyclic group
Let $G$ be a finite group, and let $M$ be a finitely generated $G$-module,
that is, a finitely generated abelian group on which $G$ acts.
We work with the first homology group
$$ H_1(G,M).$$
For any ...
- 13.4k
10
votes
1
answer
209
views
For which finite groups $G$ is $M_n(\mathbb{Q}(\zeta))$ a factor of $\mathbb{Q}[G]$?
I am cross-posting this question from my MSE post here, in case someone here can answer it.
For a finite group $G$, the rational group ring $\mathbb{Q}[G]$ has a Wedderburn decomposition:
$$
\mathbb{Q}...
- 103
2
votes
1
answer
292
views
Proving certain triangle groups are infinite
[Cross-posted from MSE]
Consider the Von Dyck group
$$ G = \langle x,y\mid x^a=y^b=(xy)^c=1\rangle $$
where $a,b,c\ge3$. Because $G$ is infinite and residually finite, it has an infinite family of ...
- 4,327
9
votes
1
answer
406
views
Growth of powers of symmetric subsets in a finite group
(This question was originally asked on Math.SE, where it was answered in the abelian case)
Let $G$ be a finite group, and let $A$ be a symmetric subset of $G$ containing the identity (i.e., $A^{-1}=A$ ...
- 2,474
2
votes
3
answers
288
views
A subgroup of index 2 for a solvable finite group transitively acting on a finite set of cardinality $2m$ where $m$ is odd
Let $G$ be a finite group acting transitively on the finite set $X=\{1,2,\dots, 2m\}$
of cardinality $2m\ge 6$ where $m$ is odd.
Question 1. Is it true that $G$ always has a subgroup $H$ of index 2
...
- 13.4k
0
votes
0
answers
83
views
The relation between two characteristic subgroups in finite p-group
Suppose $G$ is a finite $p$-group. Let
\begin{align*}
\mho_{1}(G)=\langle a^p\mid a\in G\rangle,\quad\Omega_{1}(G)=\langle a\in G\mid a^p=1\rangle.
\end{align*}
There are examples such that $|G|\leq |\...
- 71
2
votes
1
answer
259
views
Does any finite group of order $2m$ with odd $m$ have a subgroup of index 2? [closed]
Let $G$ be a finite group of order $2m$ where $m>1$ is an odd natural number.
Question. Is it true that any such $G$ has a subgroup $H$ of index 2?
If yes, I would be grateful for a reference or ...
- 13.4k
1
vote
0
answers
96
views
Closed collections of finite groups
Let $\mathcal{C}$ be a collection of (isomorphism classes of) finite groups with the following properties:
If $G\in\mathcal{C}$ and $H$ is a homomorphic image of $G$, then $H\in\mathcal{C}$
If $G\in\...
- 917
19
votes
1
answer
1k
views
Is applying Feit–Thompson’s theorem for the nonexistence of a simple group of order $1004913$ really a circular argument?
In p.212 of Dummit–Foote’s Abstract Algebra, 3rd Edition, an analysis of a hypothetical simple group $G$ of order $1004913 = 3^3 \cdot 7 \cdot 13 \cdot 409$ is carried out. The authors write:
We ...
- 301
9
votes
2
answers
525
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
$$
C_m \rtimes_k C_n ...
- 465
9
votes
1
answer
270
views
A question related to Jordan's theorem on subgroups of $\mathrm{GL}_n(\mathbb{C})$
$\newcommand{\C}{\mathbb{C}}$
$\newcommand{\mr}{\mathrm}$
For any positive integer $n$, let $f(n)$ be the minimal integer with the following
property:
For any finite subgroup $G < \mr{GL}_n(\C)$ ...
- 10.5k
2
votes
0
answers
384
views
Generalized conjugacy classes in (topological) groups
Let $G$ be a topological group. We define an equivalence relation on $G$ as follows:
For $a,b\in G$ we set $a\sim b$ if the following two maps are topologically conjugate:
$$x\mapsto ax,\qquad x\...
- 280
1
vote
0
answers
111
views
Reduction mod 2 for orthogonal groups
Setting Let $k$ be a real quadratic field, $\mathbb Z_k$ its ring of integers. Let $n$ be an even integer $A$ a symmetric $n$-by-$n$ matrix with coefficients in $\mathbb Z_k$. Let $L$ be the lattice $\...
- 3,339
2
votes
0
answers
117
views
Subgroups of a finite group whose conjugates intersect to conjugates of a specified subgroup
I have encountered a mysterious condition on finite groups in my research, and would like help understanding it better.
Let $G$ be a finite group, and let $H\leq K\leq G$ be a chain of subgroup ...
- 83
5
votes
0
answers
186
views
Subgroups of the symmetric group and binary relations
Motivation
The following came up in my work recently. (NB this is the motivation, not the question I'm asking. You can skip to the actual question below, which is self-contained, but not self-...
- 333
3
votes
0
answers
70
views
Diameters of permutation groups with transitive generators
Suppose that for a permutation puzzle on $n$ elements we have a subroutine to place any $m>3$ elements in arbitrary positions (possibly scrambling the rest). Can we solve the puzzle (if it is ...
- 5,847
4
votes
0
answers
181
views
A different approach to proving a property of finite solvable groups
Edit: I'd be happy to hear any vague thoughts you might have, however far they may be from a complete solution!
I asked this on math.stackexchange a couple of days ago, but it didn't attract any ...
- 917
7
votes
1
answer
222
views
On a generalization of Schur-Zassenhaus
Disclaimer: I'm not a group theorist, I arrived at the following question from algebraic geometry.
The first half of the Schur-Zassenhaus theorem states that, if $N$ is a normal subgroup of a finite ...
- 880
0
votes
0
answers
106
views
$G\cdot H$ with $G,H$ non-Abelian finite simple
Can a non-split extension of one non-Abelian finite simple group by another exist?
- 2,613
1
vote
0
answers
80
views
Central-by-cyclic
This is a following-up question of this.
Lemma 2.4 from Robert Griess' paper "Elementary abelian $p$-subgroups of algebraic groups" states:
(i) Let $T$ be a finite $p$-group whose Frattini ...
- 501
1
vote
1
answer
142
views
$|C(E):C(E)\cap C(Z(U))|=1$ or $p$
Lemma 2.4 from Robert Griess' paper "Elementary abelian $p$-subgroups of algebraic groups" states:
(i) Let $T$ be a finite $p$-group whose Frattini subgroup is cyclic and central. Then $T'$ ...
- 501
5
votes
1
answer
260
views
Extension of base field for modules of groups and cohomology [duplicate]
Let $G$ be a group and let $K/k$ be a field extension. Suppose that $V$ is a $kG$-module, and let $V_K = K \otimes_k V$ be the $KG$-module given by changing the base field.
Is it true that $H^n(G,V_K) ...
- 211
37
votes
2
answers
3k
views
Why does the monster group exist?
Recently, I was watching an interview that John Conway did with Numberphile. By the end of the video, Brady Haran asked John:
If you were to come back a hundred years after your death, what problem ...
- 567
1
vote
0
answers
101
views
Finite groups of prime power order containing an abelian maximal subgroup
Let $G$ be a finite $p$-group containing an abelian maximal subgroup. Then it is a well-known result that $|G:Z(G)|=p|G'|$. If in addition $G$ is of nilpotent class 2, then $|G:Z(G)|\leq p^{r+1}$, ...
4
votes
1
answer
197
views
Mackey coset decomposition formula
I have a question about following argument I found
in these notes on Mackey functors:
(2.1) LEMMA. (page 6) Let $G$ be a finite group and $J$ any subgroup. Whenever $H$ and $K$ are subgroups of $J$, ...
- 5,716
0
votes
0
answers
115
views
normalizer info for subgroups
In [1], Griess classified the maximal nontoral elementary abelian subgroups of algebraic groups. For the exceptional types, normalizer info was also given. Is there any work out there providing ...
- 501
2
votes
0
answers
62
views
Are the integer points of a simple linear algebraic group 2-generated?
Set Up:
Let $ K $ be a totally real number field. Let $ \mathcal{O}_K $ be the ring of integers of $ K $. Let $ G $ be a simple linear algebraic group. Suppose that $ G(\mathbb{R}) $ is a compact Lie ...
- 3,429
4
votes
1
answer
191
views
Projective representations of a finite abelian group
Projective representations of a group $G$ are classified by the second group cohomology $H^2(G,U(1))$. If $G$ is finite and abelian, it is isomorphic to the direct product of cyclic groups
$$
G\cong ...
- 839
9
votes
0
answers
169
views
Cyclic numbers of the form $2^n + 1$
A cyclic number (or cyclic order) is a number $m$ such that the only group of order $m$ is the cyclic group $\mathbb{Z}/m\mathbb{Z}$. The set of cyclic numbers admits a couple of cute number-theoretic ...
- 333
4
votes
1
answer
224
views
Condition on $q$ for inclusion $p^{1+2r}\cdot\operatorname{Sp}(2r,p)\leqslant \operatorname{GU}(p^r,q)$
Let $p$ be an odd prime. What's the condition on $q$ for
$$
p^{1+2r}\cdot\operatorname{Sp}(2r,p)\leqslant \operatorname{GU}(p^r,q)\;?
$$ I did some computation and seemed that $q\equiv -1$(mod $p$) ...
- 501
4
votes
1
answer
118
views
CFSG-free proof for classifying simple $K_3$-group
Let $G$ be a finite nonabelian simple group.
We call $G$ a $K_3$-group if $|G|=p^aq^br^c$ where $p,q,r$ are distinct primes and $a,b,c$ are positive integers.
My question is: Is there a CFSG-free ...
- 385
2
votes
1
answer
168
views
$\mathrm{PSL}_3(4)$ inside the Monster group
Which quasisimple groups with central quotient $G\cong\mathrm{PSL}_3(4)$ are isomorphic to subgroups of the Monster sporadic group? So far I know that $G$ itself is not and that $2\cdot G$, $2^2\cdot ...
- 2,613
2
votes
1
answer
184
views
Sparsity of q in groups PSL(2,q) that are K_4-simple
One of the problems that has come up during my research concerns $K_4$-simple groups (simple groups with $4$ prime divisors). The only (potentially) infinite family of groups satisfying this condition ...
- 23
0
votes
0
answers
65
views
Is a Lagrangian subgroup of a metric group isomorphic to its quotient?
A metric group is a finite abelian group $G$ with a quadratic function
$$q:G\rightarrow \mathbb R/\mathbb Z\;,$$
that is,
$$M(a,b):= q(a+b)-q(a)-q(b)$$
is bilinear in $a$ and $b$ [edit: and non-...
- 2,839
7
votes
2
answers
581
views
Cohomology of $\operatorname{GL}_3(\mathbb{F}_2)$
I’m wondering what is known about the cohomology of $\operatorname{GL}_3(\mathbb{Z}_2)$, the general linear group of $3\times3$ matrices over the finite field $\mathbb{Z}_2$. There are results in ...
- 397
5
votes
2
answers
218
views
Unimodality of sequence of number of subgroups in $p$-groups
It's easy to know that the sequence of number of subgroups is unimodal for elementary abelian $p$-groups. I want to know if the result is true for any $p$-group.
More, precisely, let $G$ be a finite $...
- 71
3
votes
1
answer
178
views
normalizer quotient is $\operatorname{GL}_2(p)$
Let $p$ be a prime and let $w$ be a primitive $p$-th root of unity in $\mathbb{C}$. There is an element $e$ of order $p$ in $G=\operatorname{PGL}_n(\mathbb{C})$ where $n=pk$ and
$$e=\left[\left(\begin{...
- 501
5
votes
1
answer
228
views
Product of all conjugacy classes
Related to this post of my coauthor Sebastien, I should also mention that one can also prove the following dual result:
For any finite group G, the following identity holds:
$$
\left(\prod_{j=0}^m \...
- 837
0
votes
0
answers
113
views
Comparing the perfect groups of order 1344
Take two nonisomorphic perfect groups of order 1344 and label the elements of each with the numbers 1 through 1344, then superimpose their respective Cayley tables (for simplicity’s sake, the nth row ...
- 2,613
5
votes
0
answers
273
views
Adjoint identity on finite nilpotent groups
Let $G$ be a finite nilpotent group. Consider the following (adjoint) identity from [BuPa, Theorem 9.6]:
$$\left(\prod_{\chi \in \mathrm{Irr}(G)} \frac{\chi}{\chi(1)} \right)^2 =\frac{|Z(G)|}{|G|}\...
- 25.5k