Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

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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 ...
Daniel Sebald's user avatar
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 ...
abiteofdata's user avatar
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 $|...
Dave Benson's user avatar
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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 ...
Sam Hughes's user avatar
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-...
Andi Bauer's user avatar
  • 2,901
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 ...
Noah B's user avatar
  • 403
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 $...
gdre's user avatar
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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{...
user488802's user avatar
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 ...
semisimpleton's user avatar
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 \...
Sebastian Burciu's user avatar
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 ...
Daniel Sebald's user avatar
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|}\...
Sebastien Palcoux's user avatar
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_{...
semisimpleton's user avatar
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 ...
neelkanth's user avatar
  • 141
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 $...
Yuri Bilu's user avatar
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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 ...
Dave Benson's user avatar
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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)...
Andrea Antinucci's user avatar
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 ...
user488802's user avatar
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 ...
Andrea Antinucci's user avatar
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 $\mathrm{Aut}(G)$ ...
youknowwho's user avatar
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 ...
Zach Hunter's user avatar
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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 ...
Mahtab's user avatar
  • 247
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 ...
Stefan Kohl's user avatar
  • 19.5k
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\...
Moomoo Angel line's user avatar
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 ...
user488802's user avatar
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 $\sigma$ ...
Andrea Antinucci's user avatar
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 ...
semisimpleton's user avatar
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 ...
Dan Sims's user avatar
  • 151
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 ...
semisimpleton's user avatar
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 ...
semisimpleton's user avatar
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 ...
Ian Gershon Teixeira's user avatar
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 $=$ ...
Dave Benson's user avatar
  • 11.6k
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, ...
Dac0's user avatar
  • 285
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 ...
H A Helfgott's user avatar
  • 19.3k
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 ...
semisimpleton's user avatar
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 $C_G(F)=1$). ...
user44312's user avatar
  • 509
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 ...
Steve Stahl's user avatar
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 (...
Makenzie's user avatar
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 $\...
David Roberson's user avatar
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{...
BasicQuestionBot's user avatar
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 \...
Nate Ackerman's user avatar
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 ...
stupid boy's user avatar
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 ...
Joachim König's user avatar
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 $p$...
user44312's user avatar
  • 509
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 $1$-$9$-$6$, $1$-$35$-$28$, $1$-$135$-$...
user488802's user avatar
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 ...
user1044791's user avatar
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. ...
Gillyweeds's user avatar
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 \...
Moomoo Angel line's user avatar
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 ...
Mare's user avatar
  • 25.7k
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 ...
A. Bailleul's user avatar
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