Questions tagged [finite-groups]
Questions on group theory which concern finite groups.
2,259
questions
2
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$\mathrm{PSL}_3(4)$ 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
votes
1
answer
185
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 ...
8
votes
0
answers
173
views
Singularity category of a hypersurface associated to $M_{11}$
For reasons to do with classifying spaces of finite groups, I have the following algebra. Let $k$ be a field of characteristic two, and let $R = k[x,y,z]/(x^2 y + z^2)$, as a graded $k$-algebra with $|...
4
votes
1
answer
110
views
$\mathbb{Z}$-forms of rational representation of finite group
Let $G$ be a finite group and let $\rho\colon G\to \mathrm{GL}_n(\mathbb{Q})$ be a representation of $G$.
How does one go about classifying the $\mathbb{Z}$-forms of $\rho$?
For example: here it is ...
0
votes
0
answers
75
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-...
7
votes
2
answers
597
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 ...
5
votes
2
answers
221
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 $...
3
votes
1
answer
190
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
$$e=\left[\left(\begin{...
2
votes
0
answers
84
views
Reference request: structure of group of units of finite group ring
Let $G$ be a finite group, let $F$ be a finite field and let $F[G]$ be the group algebra of $G$ over $F$.
What is known about the structure of the group of units $F[G]^\times$? Of course, it must ...
5
votes
1
answer
262
views
Product of all conjugacy classes
Related to this post of my coauthor Sebastien, I should also mention that one can also prove the following dual result:
For any finite group G, the following identity holds:
$$
\left(\prod_{j=0}^m \...
0
votes
0
answers
120
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 ...
5
votes
0
answers
344
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|}\...
3
votes
1
answer
306
views
A generalisation of induced representations
Let $G$ be a finite group, and $H\subseteq G$ a subgroup. Let $F$ be a field. Let $W$ be a finite-dimensional $F[H]$-module. Let $T$ be a left transversal of $H$ in $G$. Then we can define:
$W^G=\sum_{...
1
vote
1
answer
122
views
Example of a group algebra with commutative Jacobson radical
I am searching a simple example of a finite group $G$, so that the Jacobson radical $J(FG)$, of group algebra $FG$ is commutative, where $F$ is a finite field. I know example for that if $G$ is any ...
12
votes
1
answer
430
views
abelian quotients of permutation groups
Let $G$ be a subgroup of the permutation group $S_n$, and let $H$ be a normal subgroup of $G$ such that the quotient group $G/H$ is abelian. What is the best known upper estimate for the cardinality $...
9
votes
0
answers
114
views
Is there a strictly coassociative resolution of polynomial growth, for a finite group?
Let $G$ be a finite group and $k$ a field of characteristic $p$. It is well known, thanks to the work of Quillen, that the trivial $kG$-module $k$ has a projective resolution of polynomial growth. To ...
0
votes
1
answer
192
views
Are these two natural cohomology classes of a manifold constructed from a 1-cochain and a group extension equal?
Let $X$ be a manifold, $G$ and $A$ finite abelian groups and $\epsilon \in H^2(G,A)$ a group cohomology class (for the moment I am assuming there is no action of $G$ on $A$). Given $\alpha \in H^1(X,G)...
2
votes
2
answers
206
views
is the embedding $\mathrm{Sp}_{2m}(p)\leqslant S_{p^m-1}$ possible?
Is the following embedding possible?
$\mathrm{Sp}_{2m}(p)\leqslant S_{p^m-1}$ where $S_{p^m-1}$ is a symmetric group and $p$ is prime. I see that when $p=3$ and $m=3$, the order of the former does ...
1
vote
0
answers
120
views
Cohomology of the classifying space of a semidirect product, and some specific examples with cyclic groups
Let $G$ and $A$ be finite abelian groups and $\rho :G \rightarrow \text{Aut}(G)$ a representation of $G$. We can form the semidirect product $A\rtimes _{\rho} G$. Just to agree on the notation this is ...
8
votes
1
answer
307
views
Sylow $p$ of $\mathrm{Aut}(G)$ with $G$ finite simple?
I met the following problem:
Let $G$ be a finite simple group (non-commutative, otherwise trivial). Let $p$ be a prime number not dividing $|G|$. Prove that any Sylow $p$ subgroup of $\mathrm{Aut}(G)$ ...
3
votes
1
answer
113
views
Bounding size of group by number of generators, order of elements, and nilpotency class (Restricted Burnside's)
I have been looking at some papers on the "restricted Burnside problem". On page 4 of Vaughan-Lee and Zelmanov's survey, "Bounds in the restricted Burnside problem", I think they ...
3
votes
1
answer
195
views
Assigning a finite number, $n(G)$, to a finite group $G$ with this property
I want to assign a finite number, $n(G)$, to a finite group $G$ such that if $H$ is a proper retract of $G$, then $n(H)\lneq n(G)$. By a retract of $G$, I mean a subgroup $H$ of $G$ for which there is ...
6
votes
0
answers
189
views
Finite simple groups of "smooth" order
Given a finite group $G$, let $p(G)$ denote the largest prime factor
of the order of $G$. For the purpose of this question,
we say that the group $G$ has smooth order if its order exceeds
the order of ...
5
votes
2
answers
246
views
The property of conjugate subgroup
$\DeclareMathOperator\Syl{Syl}$$G$ is a finite group. $N\unlhd G$, $P\in \Syl_{p}(G)$, and $M$ is a proper subgroup of $P$. Suppose that for any $h\in N_G(P)$, if $MM^h=M^hM$, then $M=M^h$.
For any $g\...
1
vote
0
answers
73
views
$C_G(E)= E \times{\rm PGL}_k(q)$
Let $r$ be an odd prime and $q$ a power of a prime $p$ where $r\neq p$.
If $r^m|n$ and $q\equiv1$ (mod $r$), then $r^{1+2m}.{\rm Sp}_{2m}(r)\le{\rm GL}_n(q)$ and $Center(r^{1+2m}.{\rm Sp}_{2m}(r))\le ...
2
votes
0
answers
169
views
Decomposition of finite abelian groups of even order if there is an involution
Let $G$ be a finite abelian group and $\sigma :G\rightarrow G$ an automorphism of order two ($\sigma\circ \sigma =id_G$). Denote by $F$ and $A$ the subgroups of fixed and anti-fixed points of $\sigma$ ...
2
votes
0
answers
111
views
Status of the automorphism tower problem for finite groups
This is problem 11.123 in the Kourovka notebook:
For a given group $G$, define the following sequence
of groups: $A_1(G) = G$, $A_{i+1}(G) = \operatorname{Aut}(A_i(G))$. Does there exist a finite ...
5
votes
2
answers
243
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 ...
8
votes
2
answers
871
views
Nonisomorphic finite groups with isomorphic Sylow subgroups
The broad theme that underlies this question is: to what extent can the study of finite groups be reduced to the study of $p$-groups?
I imagine that it is possible for a pair of nonisomorphic finite ...
4
votes
0
answers
103
views
"Interpretation" of families of conjugate subgroups in a finite group
For a fixed prime $p$, the Sylow $p$-subgroups of a given finite group are all conjugate. Here are some more examples of situations in which we find that subgroups of a finite group defined by a ...
2
votes
1
answer
93
views
Image of minimal degree representation of quasisimple group unique up to conjugacy
Let $ G $ be a quasisimple finite group. Let $ d_{min} $ be the minimum dimension of a nontrivial irrep of $ G $. Must it be the case that the image of all (nontrivial) dimension $ d_{min} $ irreps of ...
7
votes
1
answer
256
views
Semi-projective complexes of modules over a finite group
Let $G$ be a finite group and $k$ a field of characteristic $p$ dividing $|G|$. A perfect complex of $kG$-modules is by definition a finite complex of finitely generated projective ($=$ injective $=$ ...
0
votes
0
answers
111
views
Four octonionic loops to identify
A loop is a quasigroup with the identity. I have to disclose that loop-theory is something outside my expertise.
I have four loops arising from octonionic elements of unitary norm that have order 16, ...
6
votes
1
answer
300
views
Number of points on a linear algebraic group over a finite field
Let $G$ be a linear algebraic group defined over a finite field $\mathbb{F}_q$ as a variety of dimension $d$. What would be a good, simple lower bound for $G(F_q)$?
One can get something fairly nice ...
9
votes
1
answer
866
views
Representations of finite groups over the "field with one element"
Have there been any attempts to extend the "F_un" analogy to the representation theory of finite groups?
If I might be allowed some speculation:
If combinatorics can be regarded as analagous ...
5
votes
1
answer
284
views
Finite abelian group admits a Frobenius group of automorphism
Suppose that a finite group $G$ admits a Frobenius group of
automorphisms $F H$ with kernel $F$ and complement $H$ such that $F$
acts without nontrivial fixed points (that is, such that $C_G(F)=1$).
...
4
votes
2
answers
283
views
A finite group whose conjugacy classes outside a normal subgroup have equal size
I would like to understand the possible structure of finite groups $G$ that has a normal subgroup $N$ of index $p$ (a prime) such that conjugacy classes of $G$ outside $N$ have equal size. Another way ...
2
votes
1
answer
150
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 (...
4
votes
0
answers
102
views
Doubly stochastic matrices that remain doubly stochastic after conjugating by the character table of a finite abelian group
I am curious if anything is known about the following.
Let $\Gamma$ be a finite abelian group, and let $\chi$ be its character table, normalized so that it is a unitary matrix. E.g., if $\Gamma$ is $\...
17
votes
5
answers
974
views
Does the map $\mathrm{SL}(n,\mathbb{Z}/p^2) \rightarrow \mathrm{SL}(n,\mathbb{Z}/p)$ split?
$\DeclareMathOperator\SL{SL}$ $\DeclareMathOperator\GL{GL}$The question is the one in the title: for a prime $p$, does the obvious surjection $\pi\colon \SL(n,\mathbb{Z}/p^2) \rightarrow \SL(n,\mathbb{...
3
votes
2
answers
248
views
Cancelable commutative monoids with finite maximal subgroups
Suppose $\mathcal{M} = (M, +, 0)$ is a cancelable commutative monoid. Let $G$ be the maximal subgroup of $M$, i.e.
$$G = \{ a \in M \colon (\exists b \in M)\, a + b = 0 \}.$$
For $a, b \in M$ say $a \...
0
votes
0
answers
168
views
A characterisation of full subgroups of $\mathrm{GL}_n(\mathbf{F}_p)$
Let $p\geq 5$ be a prime and $\mathbf{F}_p$ a finite field of characteristic $p$. A subgroup of ${\rm GL}_n(\mathbf{F}_p)$ is full if it contains ${\rm{SL}}_n(\mathbf{F}_p)$. When $n=2$, we have the ...
35
votes
1
answer
1k
views
What is the smallest group not known to be a Galois group over $\mathbb{Q}$?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$What is the smallest group not known to be a Galois group over $\mathbb{Q}$?
Variants have been asked here before (e.g. Which small finite ...
1
vote
0
answers
165
views
Irreducible module of finite simple groups
Let $G$ be a finite simple group and $p$ be a prime divisor of $|G|$.
Let $V$ be a nontrivial irreducible $\mathbb{F}_p[G]$ module.
I would like to understand the relation of $|V|$ and $|G|_p$ (the $p$...
1
vote
1
answer
135
views
Orbit sizes of $G=\operatorname{SO}^{+}_{2n}(2)$
Let $G=\operatorname{SO}^{+}_{2n}(2)$. I did some Magma computation and found there were $3$ orbits on the natural $G$-set when $n=2,3,4$. The orbit sizes are $1$-$9$-$6$, $1$-$35$-$28$, $1$-$135$-$...
6
votes
1
answer
539
views
If G is an almost simple group, then Aut(G) is complete?
If G is an almost simple group, then Aut(G) is complete?
Apologies - I meant to post this on Stack Exchange
Just wondering if anyone has a reference to the above - it's quoted on Wikipedia (so ...
3
votes
1
answer
230
views
Is there anything known about the lower central series of a group $G\wr C_p$?
Let $G$ be a $p$-group contained in $S_{p^m}$ for some $m\in \mathbb{N}$, this is, the symmetric group of degree $p^m$. Assume that the lower central series, LCS for short, of $G$ is well understood.
...
2
votes
0
answers
148
views
The maximal subgroups of a finite solvable group
$\DeclareMathOperator\Syl{Syl}$$G$ is a finite solvable group and $G=Q\rtimes(P\times R)$, where $P\in \Syl_{p}(G)$ with $P$ is cyclic, $Q\in \Syl_{q}(G)$ with $Q$ is normal elementary abelian, $R\in \...
9
votes
2
answers
655
views
Finite subgroups of $\mathrm{SO}(n)$ and $\mathrm{O}(n)$
Question 1:Is there a reference that lists all possible finite subgroups and their orders of $\mathrm{SO}(n)$ and $\mathrm{O}(n)$ for $n=4$ or even higher $n$ over the real numbers?
I can only find ...
15
votes
1
answer
692
views
Finite abelian groups with fewer automorphisms than a subgroup
It is not too hard to find examples of finite groups which have fewer automorphisms than one of their subgroups. For example $\mathcal D_4 \times \mathbb Z/2\mathbb Z$ (where $\mathcal D_4$ is the ...