Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

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6 votes
2 answers
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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 ...
Chris's user avatar
  • 163
8 votes
0 answers
376 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) \...
Ian Gershon Teixeira's user avatar
5 votes
0 answers
205 views

Extending representations of $\mathrm{GL}(n,\mathbb{F}_p) \times \mathrm{GL}(m,\mathbb{F}_p)$ to $\mathrm{GL}(n+m,\mathbb{F}_p)$

$\DeclareMathOperator\GL{GL}$In this question, representations means finite-dimensional complex representations. Fix some $n,m \geq 2$ and some prime $p$. I'm interested in representations $V$ of $\...
Annie's user avatar
  • 51
2 votes
0 answers
122 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-...
OrdinaryAnon's user avatar
1 vote
1 answer
106 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 ...
Vicent Miralles's user avatar
1 vote
1 answer
161 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 $\operatorname{cd}(G)$. ...
C. Simon's user avatar
  • 577
1 vote
0 answers
118 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 $G$. ...
stupid boy's user avatar
5 votes
2 answers
383 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 ...
David Loeffler's user avatar
3 votes
0 answers
165 views

Abelian characters and odd perfect numbers?

This question is about applications of abelian characters to odd perfect numbers: Context and Definitions: Let $n$ be a natural number and $D_n$ be the set of divisors. We can make this set to a ring ...
mathoverflowUser's user avatar
6 votes
2 answers
240 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 ...
Mikhail Borovoi's user avatar
10 votes
1 answer
216 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}...
tl981862's user avatar
  • 103
5 votes
0 answers
219 views

What is the Fourier transform in modular representation theory?

For a finite group $G$ there is the Fourier transform $\displaystyle \hat{f}(\rho)=\sum_{g \in G} f(g)\rho(g)$ with inverse $$\displaystyle f(g)=\frac{1}{|G|}\sum_{\rho}d_{\rho}\operatorname{Tr}\left(\...
Jackson Walters's user avatar
2 votes
1 answer
320 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 ...
Steve D's user avatar
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9 votes
1 answer
431 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., $A^{-1}=A$ ...
Thomas Browning's user avatar
2 votes
3 answers
297 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 ...
Mikhail Borovoi's user avatar
0 votes
0 answers
86 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 |\...
gdre's user avatar
  • 71
2 votes
1 answer
365 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 ...
Mikhail Borovoi's user avatar
1 vote
0 answers
100 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\...
semisimpleton's user avatar
0 votes
1 answer
175 views

Finite subgroups of $\mathrm{O}_n(\mathbb R)$ from finite subgroups of $\mathrm{SO}_n(\mathbb R)$

Let $G$ be a finite subgroup of $\mathrm{SO}_n(\mathbb R)$. We also assume $G$ to be "maximal" in the sense that for every $g\in\mathrm{SO}_n(\mathbb R)\setminus G$, we have that $\overline{\...
Andrea Aveni's user avatar
1 vote
0 answers
153 views

Centraliser of a finite group

Let $G=\operatorname{Sp}(8,K)$ be a symplectic algebraic group over an algebraically closed field of characteristic not $2$. We have a vector space decomposition $V_8=V_2\otimes V_4$ where the $2$-...
user488802's user avatar
20 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 ...
Kazune Takahashi's user avatar
9 votes
2 answers
602 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 ...
Jianing Song's user avatar
9 votes
1 answer
275 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 $G < \mr{GL}_n(\C)$ ...
naf's user avatar
  • 10.5k
2 votes
0 answers
406 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\...
Ali Taghavi's user avatar
1 vote
0 answers
112 views

What is $H^*(\mathbb{CP}^{2^N-1}/\Sigma_n;\mathbb{Z})$ when $N=\binom{n}{2}$?

$H^*((S^3)^N/\Sigma_n;\mathbb{Q})$ is computed here. It makes a little more sense to compute $H^*((S^2)^N/\Sigma_n;\mathbb{Q})$ given that global phase is irrelevant. The proof is exactly the same. ...
Jackson Walters's user avatar
1 vote
0 answers
114 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 $\...
Jean Raimbault's user avatar
2 votes
0 answers
121 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 ...
Chase's user avatar
  • 93
5 votes
0 answers
193 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-...
Z. A. K.'s user avatar
  • 586
3 votes
1 answer
190 views

Finite subgroup of $\operatorname{Sp}(2n,K)$

Let $G$ be the algebraic group $\operatorname{Sp}(2n, K)$ where $K$ is an algebraically closed field of characteristic not $2$. There is a quaternion subgroup $Q$ such that $Q/Z(G)$ is elementary ...
user488802's user avatar
3 votes
0 answers
78 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 ...
Dmytro Taranovsky's user avatar
4 votes
0 answers
193 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 ...
semisimpleton's user avatar
7 votes
1 answer
240 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 ...
Giulio Bresciani's user avatar
0 votes
0 answers
109 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?
Daniel Sebald's user avatar
1 vote
0 answers
84 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 ...
user488802's user avatar
1 vote
1 answer
143 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 $T'$ ...
user488802's user avatar
1 vote
0 answers
48 views

A question on width vs covering of the subgroup generated by a conjugacy class in a finite group

Let $G$ be a finite group and $C$ be a conjugacy class of $G$. It is clear that there exists $k\in \mathbb{N}$, such that $1\cup C\cup C^2 \cup \cdots \cup C^k=\langle C \rangle$. Note that $\langle C ...
Riju's user avatar
  • 430
5 votes
1 answer
272 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) ...
testaccount's user avatar
38 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 ...
Leibniz's Alien's user avatar
1 vote
0 answers
104 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 $|G:Z(G)|\leq p^{r+1}$, ...
Hamid Shahverdi's user avatar
4 votes
1 answer
254 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 $J$, ...
user267839's user avatar
  • 5,938
0 votes
0 answers
117 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 ...
user488802's user avatar
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 ...
Ian Gershon Teixeira's user avatar
4 votes
0 answers
147 views

New characters from old

(All groups in the following discussion are assumed to be finite.) Character induction is an operation that produces a character of a group given a character of a subgroup. I'm aware that there are ...
semisimpleton's user avatar
8 votes
1 answer
440 views

Trivial group cohomology induces trivial cohomology of subgroups

From the answer to another question I asked (Projective representations of a finite abelian group) and from the structure theorem of finite abelian groups it follows that if $A$ is a finite abelian ...
Andrea Antinucci's user avatar
4 votes
1 answer
222 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 ...
Andrea Antinucci's user avatar
4 votes
1 answer
206 views

Quadratic refinements of a bilinear form on finite abelian groups

$\DeclareMathOperator\Hom{Hom}$Let $A$ be a finite abelian group and $\text{Sym}(A)$ the (abelian) group of symmetric bilinear forms over $A$ valued in $\mathbb{R}/\mathbb{Z}$. A quadratic function on ...
Andrea Antinucci's user avatar
9 votes
0 answers
177 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 ...
Z. A. K.'s user avatar
  • 586
24 votes
0 answers
770 views

Revising the proof of CFSG

This is an oft-quoted excerpt from John Thompson's article "Finite Non-Solvable Groups": “... the classification of finite simple groups is an exercise in taxonomy. This is obvious to the ...
semisimpleton's user avatar
4 votes
1 answer
225 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 $p$) ...
user488802's user avatar
4 votes
1 answer
119 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 ...
user44312's user avatar
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