Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

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On $n$th class-preserving automorphism of finite $p$-group

Let $G$ be a finite non-abelian $p$-group, where $p$ is a prime. An automorphism $\alpha$ of $G$ is called an $n$th class-preserving if for each $x\in G$, there exists an element $g_x\in \gamma_n(G)$ ...
Rohit's user avatar
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5 votes
0 answers
170 views

How can I get my hands on McKay's "Finite p-Groups" lecture notes?

The notes I'm talking about are these. I emailed Peter Cameron, but he has since moved to a different university, and has no copies himself. I also emailed the school manager at Queen Mary, but they ...
Steve D's user avatar
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-2 votes
1 answer
208 views

class structure constants relation

Let $C_{j,k}^l$ ,usually called class structure constants, eg Jansen and Boon and/or JQ Chen, be the number of times the class $l$ is generated from the product of classes $j,k$ and $c_j=c_{-j}$ (a ...
Reza Vino Zmth's user avatar
4 votes
1 answer
408 views

Symmetric subgroups of simple algebraic groups over finite fields

Let $G$ be a simply connected simple algebraic group over a field $k$. Let $\theta\colon G\to G$ be an involution of $G$ over $k$ (an automorphism of order 2). Let $H=(G^\theta)^0$, the identity ...
Mikhail Borovoi's user avatar
11 votes
0 answers
520 views

Cyclic and prime factorizations of finite groups

A tuple $(A_1,\dots,A_n)$ of subsets of a finite group $G$ is called a factorization of $G$ if $G=A_1\cdots A_n$ and $|A_1|\cdots|A_n|=G$. In Cryptology factorizations of groups are known as ...
Taras Banakh's user avatar
  • 40.7k
10 votes
3 answers
543 views

Is each finite group multifactorizable?

Definition. A finite group $G$ is called multifactorizable if for any positive integer numbers $a_1,\dots,a_n$ with $a_1\cdots a_n=|G|$ there are subsets $A_1,\dots,A_n\subset G$ such that $A_1\cdots ...
Taras Banakh's user avatar
  • 40.7k
12 votes
2 answers
876 views

Factorizable groups

Definition. A finite group $G$ is factorizable if for any positive integer numbers $a,b$ with $ab=|G|$ there are subsets $A,B\subset G$ of cardinality $|A|=a$ and $|B|=b$ such that $AB=G$. Problem 1. ...
Taras Banakh's user avatar
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11 votes
1 answer
484 views

Is the norm element characteristic in modular group rings?

Let $G$ be a finite $p$- group and let $\varphi$ be an automorphism of $\mathbb{F}_pG$ as $\mathbb{F}_p$-algebras and let $n = \sum_{g\in G} g$ be the norm element. Does it follow that $\varphi(n)=n$? ...
HenrikRüping's user avatar
5 votes
2 answers
193 views

A question on UCS p-groups

A $p$-group $G$ is called a ${\it UCS}$ $p$-group if $G$ has precisely three characteristic subgroups, namely $1$, $\Phi(G)$ and $G$. Let $G$ be a finite UCS $p$-group of order $p^{2n}$ such that $\...
H.Shahsavari's user avatar
14 votes
2 answers
391 views

A finite group that has no decomposition of given cardinality

Let $a,b$ be two positive integer numbers. A group $G$ is called $a{\times}b$-decomposable if there are subsets $A,B\subset G$ of cardinality $|A|=a$ and $|B|=b$ such that $AB=G$ where $AB=\{xy:x\in A,...
Taras Banakh's user avatar
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8 votes
1 answer
1k views

Finite groups containing no subgroups of a given order or index

The classical Lagrange's Theorem says that the order of any subgroup of a finite group divides the order of the group. For abelian groups this theorem can be completed by the following simple fact: ...
Taras Banakh's user avatar
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6 votes
2 answers
704 views

Explicit computation of the Burnside ring

I would like to see explicit computations of the Burnside ring $A(G)$ when $G$ is a small Abelian group, such as $G=\mathbb{Z}/2,\mathbb{Z}/2^n,\mathbb{Z}/p^n$ where $p$ is an odd prime and $n\...
user51223's user avatar
  • 3,071
2 votes
1 answer
207 views

Groups with a maximal subgroup which is solvable

I would like to know results on the structure of a finite group $G$ which possesses a maximal subgroup $H$, with $H$ solvable. More precisely, about supplements of $H$, that is, decompositions $G=HK$ ...
Goa'uld's user avatar
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19 votes
3 answers
2k views

A character identity

This is related to my question, but it concerns a specific point of the proof of Schur's Theorem. Let $G$ be a finite group and $\chi$ an irreducible character of $G$. Is it true that $$\forall g\in ...
Denis Serre's user avatar
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0 votes
1 answer
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A question on Frobenius groups [closed]

Please change the title if needed. Let $p$ and $q$ be distinct primes and $G\cong(\underbrace{\mathbb{Z}_{q}\times\mathbb{Z}_{q}\times\dots\times\mathbb{Z}_{q}}_{n\,\,times})\rtimes\mathbb{Z}_{p}$, ...
H.Shahsavari's user avatar
6 votes
0 answers
88 views

Numbers where there is a unique group with integral character table

Call a number $n$ special in case there is a unique group of order n whose character table consists of only integers. This should be equivalent to the condition that the number of conjugacy classes ...
Mare's user avatar
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28 votes
1 answer
1k views

Number of irreducible representations of a finite group over a field of characteristic 0

Let $G$ be a finite group and $K$ a field with $\mathbb{Q} \subseteq K \subseteq \mathbb{C}$. For $K=\mathbb{C}$ the number of irreducible representations of $KG$ is equal to the number of conjugacy ...
Mare's user avatar
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14 votes
3 answers
630 views

Which partitions realise group algebras of finite groups?

Fix an algebraically closed field $K$ (maybe of characteristic zero first for simplicity, like $\mathbb{C}$). Given a partition $p=[a_1,...,a_m]$ of an integer $n$. We can identify $p$ with the ...
Mare's user avatar
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3 votes
1 answer
250 views

How large can a symmetric generating set of a finite group be?

Let $G$ be a finite group of order $n$ and let $\Delta$ be its generating set. I'll say that $\Delta$ generates $G$ symmetrically if for every permutation $\pi$ of $\Delta$ there exists $f:G\...
Pavel Madaj's user avatar
3 votes
1 answer
274 views

Littlewood Richardson Rule for general linear group over finite field

I just finished reading Green's 1955 paper on characters of general linear groups and have also been reading Macdonald's Symmetric Functions and Hall Polynomials. I see that there is a recursive ...
Sheel Stueber's user avatar
6 votes
1 answer
251 views

Subsets of a group with special property

Let $G$ be a finite group. We say a subset $A$ of $G$, $|A|=m$, is $(m,i)$-good, $m\geq 1$ and $0\leq i\leq m$, if there exist $g_A\in G$ such that we have $|gA\cap A|=m-i$. I need some groups such ...
Shahrooz's user avatar
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1 vote
0 answers
140 views

Does there exist an unramified $PSL_2(\mathbb{F}_p)$ extension of a quadratic field $K$?

Does there exist an unramified $PSL_2(\mathbb{F}_p)$ extension of a quadratic field $K$ for $p\geq 5$?
user avatar
5 votes
1 answer
355 views

Is the following variant of Shafarevich's theorem known?

Let $Q$ be a finite simple group which may be realized as the Galois group of some extension of $\mathbb{Q}$ (like for instance $PSL_2(\mathbb{F}_p)$ for $p\geq 5$, or the monster group) and let $G$ ...
user avatar
3 votes
1 answer
143 views

Bounding the derived length of a solvable group given the degrees of the irreducible monomial characters

Much is known about the derived length of a solvable group given the degrees and cardinality of the set of degrees of the irreducible characters. Martin Isaacs and Donald Passman pretty much started ...
Joakim Færgeman's user avatar
3 votes
1 answer
864 views

Conditions for a solvable group to have a non-trivial center

I am working on a problem in character theory where I try to bound the derived length of a solvable group using information about its characters. In my specific case, it will be extremely helpful for ...
Joakim Færgeman's user avatar
3 votes
0 answers
59 views

Zero divisors with support size 3 in complex group algebras of residually finite groups

Conjecture. There exists a function $f:\mathbb{N} \rightarrow \mathbb{N}$ such that if $\beta$ is a non-zero element of the complex group algebra $\mathbb{C}[G]$ of a finite group $G$ such that $1\...
Alireza Abdollahi's user avatar
6 votes
1 answer
345 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 $G$ ...
Alireza Abdollahi's user avatar
10 votes
1 answer
241 views

Low dimensional representations of $SL_n(\mathbb{Z}/p^\ell \mathbb{Z})$

When $\ell = 1$ I know that the smallest non-trivial irreducible complex representations of $SL_n(\mathbb{Z}/p\mathbb{Z})$ has dimension $\frac{p^n - 1}{p-1} - 1$ (with maybe some exceptions for ...
Nate's user avatar
  • 1,992
6 votes
0 answers
206 views

Examples for Bogomolov multiplier of finite group

Take $G$ to be a finite group. The projective representations of $G$ are classified by the group cohomology $H^2(G,U(1))$. Here $G$ has a trivial action on $U(1)$. We focus on the restriction map $H^...
Chao-Ming Jian's user avatar
1 vote
0 answers
90 views

What is an upper limit of relative size of conjugacy class of the transitive finite group?

What is $$ \limsup_{n\to\infty} \sup_{\deg(G)=n} \left( \frac{\max_{x\in G} \left|\operatorname{conj}(x)\right|}{\operatorname{ord}(G)}\right),$$ $G$ transitive permutation group? And what are the ...
Slepecky Mamut's user avatar
21 votes
2 answers
667 views

Does $\mathrm{SL}_{n}(\mathbb{Z}/p^{2})$ have the same number of conjugacy classes as $\mathrm{SL}_{n}(\mathbb{F}_{p}[t]/t^{2})$?

Let $p$ be a prime; $\mathbb{F}_{p}$ is the field with $p$ elements and $\mathbb{F}_{p}[t]$ the ring of polynomials in $t$ over $\mathbb{F}_{p}$. Does $\mathrm{SL}_{n}(\mathbb{Z}/p^{2})$ have the ...
A Stasinski's user avatar
  • 3,743
3 votes
1 answer
150 views

On decomposition of finite Abelian groups

It is easy to see that for any finite Abelian group $G$ and any numbers $a,b$ with $|G|=ab$ there exist a subgroup $A\subset G$ and a subset $B\subset G$ such that $|A|=a$, $|B|=b$ and $G=A+B$, where $...
Taras Banakh's user avatar
  • 40.7k
6 votes
1 answer
254 views

Reference request: an elementary result on characters of finite abelian groups

The referee of a paper I submitted to a journal asked me to include a reference of the following elementary result on characters of finite abelian groups: Let $A$ be a finite abelian group of order $...
efs's user avatar
  • 3,089
5 votes
1 answer
772 views

Determining the conjugacy classes of a wreath product $G \wr S_n$

If $G$ is a finite group and its conjugacy classes are known, can the conjugacy classes of the wreath product $G \wr S_n \cong G^n \rtimes S_n$ be determined?
Chris Russell's user avatar
5 votes
1 answer
142 views

Is there a subgroup of dual depth 3?

This post is motivated by an exchange with Zhengwei Liu. It is more than the dual version of this post, because we consider any subgroup (instead of just maximal), and even more at the end... Let's ...
Sebastien Palcoux's user avatar
11 votes
2 answers
555 views

About normal minimal subgroups not in the Frattini

In Neukirch--Schmidt--Wingberg, "Cohomology of Number Fields", Second edition, page 624, Exercise 2, it is stated the following fact. $\textbf{Claim}$: If $N$ is a normal subgroup, minimal among ...
Tom1909's user avatar
  • 113
6 votes
3 answers
342 views

Is there a maximal subgroup of depth 3?

Let's first define what we mean by depth of a subgroup. Let $G$ be a finite group and $H$ a subgroup. Let $(V_i)_{i \in I}$ and $(W_j)_{j \in J}$ be the irreducible complex representations of $G$ ...
Sebastien Palcoux's user avatar
4 votes
0 answers
411 views

Normalizer of a split torus

Let $G$ be a connected reductive group split over a field $k$. Let $T$ be a maximal split torus of $G$. Consider $N_G(T)$, the normalizer of $T$ in $G$, we have $N_G(T)/T \cong W$, the Weyl group of $...
Shawn's user avatar
  • 453
7 votes
0 answers
300 views

Number of elementary abelian subgroups of extraspecial $2$-groups

Let $G$ be an extraspecial $2$-group, i.e. $Z(G)=G'=\Phi(G)$ has order $2$. Then $|G|=2^{2n+1}$ for some $n\geq 1$, $G\cong D_8^{*n}$ or $G\cong Q_8*D_8^{*(n-1)}$, and $G$ has one of the following ...
Marius Tarnauceanu's user avatar
5 votes
2 answers
341 views

Is there a size 2 generating set of the signed symmetric group $B_n$?

The signed symmetric group $B_n$ is a permutation group where the underlying set is $B_n=\{\sigma \in S_{A_n}| \forall x \in A_n, \sigma(-x)= -\sigma(x)\}$ with $A_n=\{-n,-(n-1),-(n-2),\cdots,-1,1,\...
RTK's user avatar
  • 153
24 votes
2 answers
683 views

What's the maximum probability of associativity for triples in a nonassociative loop?

In a finite nonabelian group, the probability that two randomly chosen elements commute cannot exceed 5/8. One easy proof also makes it easy to find the smallest groups that attain this bound, namely ...
John Baez's user avatar
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5 votes
1 answer
489 views

A name for a group with finite abelization?

Let us recall that a group $G$ is called perfect if it coincides with its commutator subgroup $G'$, or equivalently, if its abelianization $G/G'$ is trivial. Question. Is there any name for a group ...
Taras Banakh's user avatar
  • 40.7k
7 votes
1 answer
507 views

The probability that two elements of a finite nonabelian simple group commute

It is mentioned in here (last paragraph of the first page) that Dixon proved the following result: the probability that two elements of a finite nonabelian simple group commute is at most $\frac{1}{12}...
user129021's user avatar
5 votes
0 answers
154 views

lifting of idempotents in group ring

Let $G$ be a finite group, and let $\pi:G\to Q$ be a surjective group homomophism. The map $\pi:G\to Q$ does not necessarily split, but we can always find a set theoretical splitting $s:Q\to G$. In ...
Ehud Meir's user avatar
  • 4,969
6 votes
3 answers
210 views

Groups whose poset of direct factors are lattices

Let $G$ be a finite group. Denote by $\mathcal{N}(G)$ the modular lattice of normal subgroups of $G$ and denote by $\mathcal{D}(G)$ the subposet of $\mathcal{N}(G)$ whose elements are the direct ...
Rajkarov's user avatar
  • 933
2 votes
1 answer
199 views

Extensions of lattices

Let $G$ be a finite group and $\phi:M\to N$ be a surjective homomorphism of $G$-lattices (i.e. finitely generated $\mathbb{Z}[G]$-modules, free as $\mathbb Z$-modules), with kernel $K$. For every $n\...
Tetawo's user avatar
  • 23
3 votes
1 answer
463 views

A permutation with reflection property

Consider permutations $\pi$ of the set $\{1,\dots,n\}$ having the symmetry property $\pi \pi^* \pi = \pi^*$, where $\pi^*$ is the "reflection" $k \mapsto n+1-k$. Are there references or other ...
Dirk Werner's user avatar
  • 1,838
13 votes
1 answer
739 views

How nearly abelian are nilpotent groups?

It is not uncommon to read that "nilpotent groups are 'close to abelian'."1,2 Can this sentiment be made precise in the sense of the Turán and Erdős definition of "the probability that two elements of ...
Joseph O'Rourke's user avatar
3 votes
1 answer
205 views

Operation of a p'-group on a set of p-power order and fix points

The question is related to Taft's Theorem about G-invariant radical complements. Let $A$ be an associative unitary finite-dimensional $K$-Algebra posessing a separable factor Algebra by ist nilradical....
Sven Wirsing's user avatar
4 votes
1 answer
386 views

a question on Deligne-Lusztig characters

Let $k$ be a finite field and $\bar k$ be its algebraic closure, and $F$ be the Frobenius map. Let $G$ be a reductive group over $\bar k$, $T$ be an $F$-invariant maximal torus of $G$, and $\theta$ be ...
Q. Zhang's user avatar
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