Questions tagged [finite-groups]
Questions on group theory which concern finite groups.
2,475 questions
0 votes
0 answers
42 views
What are the current open problems concerning string C-groups and regular abstract polytopes?
I have been studying the correspondence between string C-groups and regular abstract polytopes, as developed in the works of Egon Schulte, Asia Weiss, and others. A string C-group is a group generated ...
- 1
8 votes
2 answers
671 views
Hyperelliptic curve with octahedral symmetry
The hyperelliptic curve (Riemann surface) $y^2=x^8+14x^4+1$ of genus $3$ has binary octahedral symmetry. The earliest mention of this curve we found is Rodríguez, Rubí E.; González-Aguilera, Víctor. ...
- 17.7k
16 votes
4 answers
1k views
References for Burnside's theorem without character theory
I'm trying to find a textbook reference for the pure group theory proof of Burnside's $p^a q^b$ theorem, and it's surprisingly difficult to locate one. Surely there must be treatments of the ...
- 169
3 votes
0 answers
184 views
How does $\text{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ act on $\text{Irr}(G)$?
Let $G$ be a finite group. For a field $F$ (algebraically closed of characteristic $0$), let $\text{Irr}_F(G)$ denote the irreducible characters of $G$ over $F$. $\text{Gal}(\mathbb{C/R})$ acts on $\...
- 2,131
3 votes
0 answers
131 views
On number of subgroups of finite non-abelian simple groups
It is known that there exist non-isomorphic non-abelian finite simple groups with same order. For example one can refer to: Non-isomorphic finite simple groups My question is: Can there be two non-...
- 341
4 votes
1 answer
121 views
How can one obtain an inclusion of an induced module and the cokernel thereof with MAGMA?
I would like to ask a MAGMA question. In the MAGMA code below, ...
- 317
3 votes
0 answers
84 views
Subgroup structure of $\mathrm{J}_4$
Up to isomorphism, there are two groups which are maximal subgroups of both of the simple groups $\mathrm{M}_{24}$ and $\mathrm{L}_5(2)$ (using ATLAS notation). These have structure $2^4:\mathrm{A}_8$ ...
- 3,203
10 votes
0 answers
156 views
A query regarding maximal subgroups of a finite non-solvable group
This is some kind of continuation of an earlier MO post: Existence of maximal subgroups of even order which are not normal It appeared in a comment in the above post. I believe the following is true: ...
- 341
4 votes
1 answer
232 views
RT-structures in finite groups
During my research in Algebraic Geometry, I was led to the following problem in Combinatorial Group Theory, strictly related to finite quotients of pure surface braid groups. Let $G$ be a finite group....
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1 vote
1 answer
111 views
Composition factors of induced representations of semi-direct products
As I have not yet recieved any answers, this question is cross-posted from stack exchange Let $H$ be a subgroup of a finite group $G$ and let $\phi:\mathbb{Z}/2\to \text{Aut}(G)$ such that $\phi(1)(H)=...
5 votes
1 answer
318 views
Existence of maximal subgroups of even order which are not normal
Let $G$ be a finite non-solvable group. Does $G$ always have a maximal subgroup of even order which is not normal in G? Attempt: As $G$ is non-solvable, $|G|$ is even and has an element of order $2$, ...
- 341
2 votes
0 answers
110 views
Size of Chowla sets
Definition (Chowla subset). A nonempty subset $S$ of a group $G$ is called a Chowla subset if every element of $S$ has order strictly larger than $|S|$, i.e. $$\mathrm{ord}(x) > |S| \quad \text{for ...
- 379
4 votes
0 answers
154 views
Almost unipotent characters
Let us consider a split adjoint simple group $G$ over $\overline{\mathbb{F}_q}$. Then we have a Frobenius map $F$, and we can consider a finite group of Lie type, $G^F$. (We can assume that the ...
- 315
3 votes
2 answers
286 views
Examples of representations of finite groups with conditions on the tensor square
I would like to have examples of a finite group, $G$, with a finite dimensional representation, $V$, (over the complex numbers, say) with four conditions: $V$ has a $G$-invariant inner product The ...
- 570
3 votes
1 answer
537 views
A "discriminant" for finite groups
Let $G$ be a finite group, let $H\subseteq G$ be a subgroup, and let $T$ be a set of representatives for the left cosets of $H$ in $G$. Let $\lambda\in\text{Irr}(H/H')$ be a linear character of $H$. ...
- 2,131
16 votes
1 answer
589 views
When is the ring of integers of a character field the ‘character ring’?
Let $G$ be a finite group with an irreducible complex character $\chi$. Let $\mathbb Q(\chi)$ denote the field extension of the rationals generated by the values of $\chi(g)$ for $g \in G$. A theorem ...
- 333
3 votes
0 answers
164 views
When does an $FG$-module have a projective $R$-form?
Let $(F,R,k)$ be a splitting $p$-modular system for a finite group $G$. (Here, $R$ is a discrete valuation ring with residue field $k$ of characteristic $p$ and field of fractions $F$.) Let $U$ be an $...
- 2,131
5 votes
0 answers
136 views
Proof that the unitary associator isomorphism is involutive when the first and last object coincide
Consider the category of finite dimensional unitary representations of some compact group $ G $. As described here we can define a map $ \Phi_{i,j}^{k,m} $. Now suppose that $ V_i \cong V_k $ (in ...
- 4,585
13 votes
3 answers
897 views
Interesting examples of non-isomorphic groups where the probability distribution of the number of square roots of elements is the same
Given a finite group $G$, let $r(g)$ denote the number of square roots of $g$ in $G$, namely: $$ r(g) = \#\{x\in G \mid x^2 = g\}. $$ When $g$ is sampled uniformly at random from $G$, $r(g)$ becomes a ...
- 5,884
1 vote
0 answers
119 views
Cartan matrices and defect groups of blocks of group algebras
Let $G$ be a finite group, $k$ a characteristic $p$ field, and $B$ a block of the group algebra $kG$ with defect group $D$. If $k$ is sufficiently large (e.g., contains $|G|$-th roots of unity), it is ...
- 135
5 votes
2 answers
334 views
Sum over all elements in conjugacy class of $S_n$ has all integer eigenvalues in any representation?
Sorry, for the question being obvious or well-known for some, just want to reconfirm not to mislead myself and colleagues. It seems the answer might follow from previous posts by N.Elkies and B....
- 26.3k
1 vote
2 answers
434 views
Largest 3-zero-sum-free subset in $(\mathbb{Z}/4\mathbb{Z})^n$?
I’m investigating the largest subset $H \subseteq (\mathbb{Z}/4\mathbb{Z})^n$ with no three distinct vectors $x, y, z \in H$ such that $x + y + z \equiv 0 \pmod{4}$ (pointwise addition), as posed by ...
- 11
30 votes
2 answers
884 views
$\{p\text{-th powers}\} \cap \{q\text{-th powers}\} = \{pq\text{-th powers}\}$ for $p,q$ coprime in a group?
Let $G$ be a finite group, and $p$ and $q$ two coprime positive integers. Let $x \in G$. Assume that $x = y^p = z^q$ for some $y, z \in G$. Is it true that $x = w^{pq}$ for some $w \in G$ ? Quite ...
- 35.5k
2 votes
1 answer
123 views
2-periodic resolutions of $C_p$-modules
Let $p$ be a prime and denote by $C_p$ the group with $p$ elements. It is well-known that for any $C_p$-module $M$, the cohomology groups $\textrm{Ext}_{\mathbb ZC_p}^*(\mathbb Z,M)$ are 2-periodic in ...
- 135
5 votes
1 answer
178 views
Reference request: $p$-local Frobenius complements in finite groups
Let $G$ be a finite group, and let $p$ be a prime. Let $H\subseteq G$ be a subgroup, where $p$ divides $|H|$. We shall say that $H$ is a $p$-local Frobenius complement if $H\cap H^x$ is a $p'$-group ...
- 2,131
5 votes
1 answer
126 views
Do different $R$-forms for the same simple $FG$-module have the same vertex?
Let $G$ be a finite group. Let $R$ be a discrete valuation ring with residue field $k$, where $k$ has positive characteristic $p$. Let $F$ be the field of fractions of $R$. Let $V$ be a simple $FG$-...
- 2,131
4 votes
0 answers
222 views
A lower bound on the number of involutions in nonabelian finite simple groups
As a consequence of the Brauer-Fowler theorem, I am aware of an upper bound on the number of involutions in a nonabelian finite simple group G. Is there any such known lower bound on the number of ...
- 41
9 votes
5 answers
965 views
Does an irreducible representation $\;p:G\rightarrow (V \rightarrow V)$ always span the whole space of maps $V\rightarrow V$?
EDIT(Andy Putman): Since it's written in what I think is a confusing way, I'm going to rewrite the question in a different language. The original question is below. Let $G$ be a group and let $V$ be ...
- 269
5 votes
0 answers
132 views
Strange metrics on finite groups
Let $G$ be a finite group, and fix a prime $p$. For conjugate elements $a,b\in G$, define: $$d_c(a,b)=\min\{\nu_p(|H|)\mid H\triangleleft G,\text{ there is }h\in H\text{ such that }a^h=b\}$$ where, as ...
- 2,131
6 votes
1 answer
428 views
Cup-square obstructions via $j_*: H^k(G, \mathbb{F}_2) \to H^k(G, \mathbb{C}^*) $ for finite groups
Let $G$ be a finite group. Consider the homomorphism $j: \mathbb{F}_2 \to \mathbb{C}^*$ given by $j(1) = -1$. This induces a map in cohomology: $$ j_*: H^k(G, \mathbb{F}_2) \to H^k(G, \mathbb{C}^*). $$...
- 1,858
0 votes
0 answers
147 views
Finite groups with special centralizers
Let $G$ be a finite non-abelian group and $x$ is an arbitrary non-central element of $G$. Assume that for every power of $x$, say $x^k$, such that $x^k$ is not central element of $G$, we have $C_G(x)=...
- 315
0 votes
0 answers
187 views
Can every finite group be reduced to an abelian subgroup via a chain of large subgroups, under a product-closed hierarchy of group classes?
Let $P_0$ denote the class of all finite abelian groups. Motivated by the failure of the following idea: It is known (e.g., Pyber 1997) that for every finite group $G$ we can find an abelian subgroup $...
- 613
1 vote
0 answers
106 views
Subgroups of the symmetric group $S_n$ without $2^k$-type of cycles for $1\leq k\leq t$ for some given $t\leq \lfloor \frac{n}{2}\rfloor$
Let $S_n$ be a symmetric group of degree $n\in \mathbb{N}$, and $t$ a fixed positive integer with $t\leq \lfloor \frac{n}{2}\rfloor$. Question: Can we characterize the maximal subgroups of $S_n$ ...
- 643
2 votes
0 answers
151 views
Does this construction always coincide with the transfer?
(In what follows, I will read from left to right, so groups act on the right.) Let $G$ be a finite group, and let $H$ be a subgroup of $G$. Let $t_1, \ldots , t_{[G:H]}$ be a set of right $H$-coset ...
- 3,725
3 votes
2 answers
307 views
On sections from a quotient of a finite algebraic group scheme over a characteristic $0$ field
Over a field of characteristic $0$, consider a finite group scheme $G$ and its quotient by a subgroup scheme $H$. Does there exist a schematic section of the morphism from $G$ to its quotient $G\over ...
20 votes
3 answers
1k views
Are there interesting finite groups which are not small?
I was playing around with finite groups recently, and a thesis (not really a conjecture, since it's rather informal) came to my mind that "all interesting behaviour of finite groups happens ...
- 1,732
0 votes
1 answer
203 views
Complex roots of subgroup-representation polynomials
For a finite group $G$, define $$\zeta(G) := \frac{1}{|G|} \sum_{\chi \in \text{Irr}(G)} \chi(1)^3$$ where the sum is over all irreducible complex characters and $\chi(1)$ is the degree. Define the ...
1 vote
0 answers
107 views
Universal Picard groups for equivariant module categories over varying characteristics
Let $G$ be a finite group. For each prime $p$ dividing $|G|$ and field $k$ of characteristic $p$, one can consider the stable $\infty$-category of modules over the constant Mackey functor $Hk$ in $G$-...
7 votes
1 answer
402 views
A theorem of Moorhouse on (finite) groups and (simple) graphs
Given a group $G$, let $\Gamma(G)$ be the simple graph with vertex set $G \times G$, in which two distinct vertices $(x, y)$ and $(u, v)$ are adjacent if and only if either $x = u$, or $y = v$, or $xy ...
- 11.4k
0 votes
0 answers
355 views
A conjecture for factorization of group-cohomology classes
Some arguments inspired by physics (in particular a very influential paper by Wang–Wen–Witten Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions on symmetry preserving gapped ...
- 820
7 votes
0 answers
218 views
Define a finite subgroup of a Lie group by minimising a function
The Thomson problem on the $S^2$ sphere asks what configuration(s) of $N$ points minimize a particular function which is symmetric under all permutations of its arguments, $$F(x_1,\ldots,x_N) = \sum_{...
- 1,426
1 vote
0 answers
96 views
Rank of tensors with a $G-$action
I've come across this question while trying to prove a divisibility criterion for the rank of symmetric tensors. The problem can be stated in much more generality, and can be generalized as follows: ...
- 1,353
1 vote
1 answer
160 views
Clarification on the definition of $P(m,n)$ in Mann–Martínez (1996) "The exponent of finite groups"
In their 1996 paper "The exponent of finite groups", Mann and Martínez define a function $P(m,n)$ as: $$P(m,n)=\text{max}\left(\frac{R(m,n^2)}{R(m,n^2)+1},1-\frac{1-P_G(n)}{R(m,n)}\right)$$ ...
0 votes
0 answers
60 views
Generating subcycles in second order recursive dynamics on the symmetric group
This question is a concrete example of open questions I have on second order recursive dynamics on finite groups as asked in Second order recursive dynamics on finite groups . Let $S_n$ be the ...
- 271
9 votes
1 answer
278 views
Is every finite subgroup of a simple Lie group contained in a lattice?
For any simple Lie group $\mathrm{G}$ with a finite subgroup $\mathrm{H}$, does there exist a lattice $\Gamma$ with $\mathrm{H}\subseteq\Gamma\subset\mathrm{G}$?
- 3,203
1 vote
0 answers
107 views
Second order recursive dynamics on finite groups
I am looking mostly for literature and introductory material on the following subject. Consider a finite group $G$ with operation $\circ$ and the following recursive dynamics on $G$ with inital values ...
- 271
8 votes
1 answer
252 views
What is geometrically special about an orthogonal lattice (and its $GL(n,{\mathbb Z})$ copies) among all integer lattices?
Let's assume we have a lattice $L \subset {\mathbb R}^n$, given by basis vectors $v^1,v^2,\ldots,v^n$, so $L = \{ \sum_{i=1}^n k_i v^i | k_i \in \mathbb Z \}$. Let's collect all basis vectors into a ...
- 1,426
8 votes
1 answer
624 views
Congruence mod four of the number of subgroups of a finite $2$-group
This is a follow-up to my question How many subgroups can there be in a group of order 512? Is it always 2 mod 4?, which in turn was inspired by the question What finite groups have as many elements ...
- 22.4k
12 votes
1 answer
1k views
How many subgroups can there be in a group of order 512? Is it always 2 mod 4?
Some experimentation leads me to suspect that if $G$ is a finite group with $|G|=512$ then the number of subgroups of $G$ is congruent to $2$ modulo $4$. Does anyone see a good reason why this might ...
- 22.4k
4 votes
0 answers
213 views
Name/applications for a normal subgroup containing all elements of prime order
Let $G$ be a finite group, and let $\Pi G$ denote the set of all elements of prime order. A subgroup $N\leq G$ contains $\Pi G$ if and only if it satisfies $\Pi N=\Pi G$, if and only if every ...
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