Questions tagged [finite-groups]
Questions on group theory which concern finite groups.
637
questions with no upvoted or accepted answers
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An abstract zero-sum problem
I would like to know whether the problem described below has appeared in the literature and/or whether similar questions have been studied. I would be very happy to find some references or, if none ...
- 1,169
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380
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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) \...
- 3,877
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175
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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 $|...
- 11.8k
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215
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Alternate proof in Fulton–Harris of representation theoretic version of Littlewood–Richardson rule
$\DeclareMathOperator\Ind{Ind}$Let $d = d_1 + d_2$ with $d_1$, $d_2$ positive integers. Let $\lambda$ be a partition of $d_1$ and $\mu$ a partition of $d_2$, so that the Young symmetrizer construction ...
- 229
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123
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What are the stable cohomology classes of the "orthogonal groups" of finite abelian groups?
Let $A$ be a finite abelian group, and equip it with a nondegenerate symmetric bilinear form $\langle,\rangle : A \times A \to \mathrm{U}(1)$. Then you can reasonably talk about the "orthogonal ...
- 53.1k
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232
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Is there a finite group with nontrivial $H^2$ but vanishing $H^4$, $H^5$, and $H^6$?
Is there a finite group $G$ such that the group cohomology $\mathrm{H}^2_{\mathrm{gp}}(G; \mathbb{Z}/2)$ is nontrivial but $\mathrm{H}^4_{\mathrm{gp}}(G; \mathbb{Z}/2)$, $\mathrm{H}^5_{\mathrm{gp}}(G;...
- 53.1k
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240
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When does a semisimple $\mathbb{C}$-algebra come from a group?
Let $\mathcal{A}$ be a semisimple $\mathbb{C}$-algebra. By the Artin-Wedderburn theorem, it is isomorphic to a direct product of matrix algebras:
$$ \mathcal{A} = \prod_{i=1}^m M_{n_i}(\mathbb{C})$$
...
- 267
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235
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Computing the number of elementary abelian p-subgroups of rank 2 in $GL_{n}(\mathbb{F}_{p})$
Let $p$ be a prime number, and let $\mathbb{F}_{p}$ be a finite
field of order $p$. Let $GL_{n}(\mathbb{F}_{p})$ denote the general linear
group and $U_{n}$ denote the unitriangular group of $n\times ...
- 461
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147
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Finite groups with unbounded commutator width and no nontrivial central chief factor
The commutator width of a group is the smallest $n$ such that every product of commutators is a product of $n$ commutators.
My initial question was:
Do there exist finite perfect groups with ...
- 60.4k
8
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186
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Non-zero group determinant for symmetric group
Let $G$ be a finite group. Given complex numbers $x=\{x_g: g\in G\}$, one can define a $|G|\times |G|$ matrix $X$, with entries $X_{g,h} = x_{gh^{-1}}$.
Let's consider $G$ being the symmetric group $...
- 1,829
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194
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Relationship between the p-radical subgroups and the parabolics in a BN-pair generality
A theorem of Quillen says that if $G$ is a finite Chevalley group over characteristic $p$, then the poset $\mathcal{A}_p(G)$ of nontrivial elementary abelian subgroups of $G$ is homotopy equivalent (I ...
- 1,596
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271
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Integral representations of finite groups and lattice point geometry
See the edit at the bottom (April 2021)
This contains both a reference request, and a specific problem.
Let $K$ be a finite group, and let $\theta: K \to {\rm GL}(d,{\bf Z})$ be a (faithful) group ...
- 4,669
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651
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What classifies involutive automorphisms on finite groups? What classifies involutions on finite based rings?
Groups
Let $G$ be a finite group. An involutive automorphism on $G$ is an automorphism $i\colon G \to G$ such that $i^2 = 1_G$.
Question 1. What classifies involutive automorphisms on a given (non-...
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Is recognizing if a Latin square is isotopic to its transpose more efficient than computing its symmetry group?
Ihrig and Ihrig (2007) described a mathematical method for determining if a Latin square is isotopic to its transpose (where isotopic Latin squares vary by permuting the rows, columns and symbols). ...
- 1,327
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434
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A relation between intersection and product on Boolean interval of finite groups
Let $[H,G]$ be a Boolean interval of finite groups (i.e. the lattice of intermediate subgroups $H \subseteq K \subseteq G$, is Boolean). For any element $K \in [H,G]$, let $K^{\complement}$ be its ...
- 25.9k
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641
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Approximating Lie groups by finite groups
How can one approximate compact Lie groups by finite groups?
My wish is something like this:
Let $G$ be a compact Lie group.
There is a sequence of nested finite subgroups $G_n$ so that $G_n\to G$...
- 8,046
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299
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A strong sum-product "for translates" in finite fields
In the course of some recent research, I've sketched out a proof of the following result. My basis question is: is the result interesting?
Proposition There exists an absolute constant $c$ such ...
- 11.2k
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397
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Connection between two theorems on character values?
In a recent arXiv preprint here, Dipendra Prasad has revisited a 1976 theorem of Kostant (Theorem 2 in the paper On Macdonald's $\eta$-function formula, the Laplacian and generalized exponents, ...
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Outer group automorphisms preserving conjugacy classes of pairs of commuting elements
The following problem appears in the study of braided autoequivalences of representation categories of Drinfeld doubles of finite groups.
Let $G$ be a finite group. An outer automorphism $\alpha$ of ...
- 629
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899
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Jordan's Theorem on primitive permutation groups
The theorem I am referring to in the title is this:
Theorem. If $p$ is a prime and $n$ is an integer with $n \geq p+3$, then the only primitive permutation groups on $n$ points containing a $p$-...
- 3,585
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192
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Growth of spheres in FINITE nilpotent groups - Gaussian approximation (central limit theorem)?
Standard setup. Consider a group and choose generators. Word-metric (or in the other words - distance on the Cayley graph of the group+generators) - converts a group into a metric space, which is ...
- 23.9k
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115
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Endo reversible words
Let $w$ be a word in free group $F$ on finitely many generators. We will look at $w$ as word map on groups. It is clear that there exists an endomorphism $\phi$ of $F$ such that $\phi(w) = w^{-1}$ if ...
- 273
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137
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When is $\operatorname{Out}(G)$ isomorphic to the symmetries of the character table of $G$?
$\DeclareMathOperator\Out{Out}\DeclareMathOperator\CTS{CTS}$The outer automorphism group $\Out(G)$ of a finite group $G$ act on the character table by permuting columns (conjugacy classes) and rows (...
- 141
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168
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Finite group with exactly one class each of two given groups
Given two nonisomorphic finite groups, is it always possible to construct a larger finite group with exactly one conjugacy class of subgroups isomorphic to each?
- 2,689
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215
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Classification of octonionic reflection groups
I know that there exist classification theorems for real, complex, and quaternionic, reflection groups.
There are presentations for the real reflection groups, as well as further presentations for the ...
- 113
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140
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Intersection of $\mathrm{PGL}_2(q_0)$'s in $\mathrm{PGL}_2(q_0^3)$
$\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\GL{GL}$I would like an explanation for some strange numerology which I encountered when studying intersections of subfield subgroups in $\PGL_2(q)$. ...
- 11.2k
7
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419
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Are these two kernels isomorphic groups?
We have a finitely presented, infinite group $\mathsf{B}$,
coming from a geometric topology problem (it is the quotient of a braid group for a genus 2 surface).
It is generated by elements
\begin{...
- 65.1k
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103
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Irreducible representations and Jaccard Kernel for Groups?
Since each group $G$ can be considered as a subgroup of the symmetric group, we might see the elements of $G$ as permutations $\pi$.
Consider for each $\pi \in G$ the set:
$$X(\pi) := \{ (i,\pi(i)) |...
user6671
7
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172
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Finite groups such that non-central, commuting elements have the same stabilizer
Let me say that a finite, non-abelian group $G$ has the property $(P)$ if
for any two elements $x, \, y \in G - Z(G)$ such that $y \in C_G(x)$, one has $C_G(y)=C_G(x)$.
For instance, it is easy ...
- 65.1k
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242
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Simplicial model for $\mathcal{L}BG//S^1$ for a finite group $G$
$\require{AMScd}$For $X$ a (nice enough) topological space, the free loop space $\mathcal{L}X$ is the space of continuous maps from $S^1$ to $X$. This space has a natural $S^1$ action given by ...
- 6,719
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How can I get my hands on McKay's "Finite p-groups" lecture notes?
How can we find Susan 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 ...
- 4,365
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300
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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 ...
7
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139
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Small modules over finite group with large cohomology
Looking at this Example of group cohomology not annihilated by exponent of $G$? I stumbled upon one question I couldn't solve (probably because it's hard), so I post it here.
Using Lyndon resolvent, ...
- 4,436
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225
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Computation of $H^2(S_n,\mathbb{Z}/2\mathbb{Z})$
Do you have a nice modern reference where I could find the computation of $H^2(S_n,\mathbb{Z}/2\mathbb{Z})$, where the action is trivial ?
I have looked at the very few books on cohomology of groups ...
- 1,661
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132
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Asymptotic hyperbolicity constant of symmetric group
Let $\delta(G)$ be the smallest Gromov-hyperbolicity constant of a finite group $G$ (e.g., for any generating set of $G$, all triangles in the corresponding Cayley graph are $\delta(G)$-thin). Let $...
- 361
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Triviality of Semisimple Hopf Algebras of Cyclic Dimension
A cyclic number is a natural number $n$ such that any group of order $n$ is cyclic. A003277
Theorem (T. Szele, 1947): A number $n$ is cyclic if and only if it is coprime to its Euler totient $\varphi(...
- 25.9k
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Can a finite group have its third cohomology determined by a quotient group?
Does there exist a finite group $G$ satisfying either of the following?
$G$ admits a non-trivial, proper, central subgroup $A$ such that every (normalized) 3-cocycle of $G$ is cohomologous to a (...
- 802
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190
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Restriction of $H^3(M_{24}, U(1))$ to $M_{12} \rtimes \mathbb{Z}_2$
$M_{12}\rtimes \mathbb{Z}_2$ is a maximal subgroup of $M_{24}$, where $M_{24}$ and $M_{12}$ are Mathieu groups . Also, it is known that $H^3(M_{24}, U(1)) \cong \mathbb{Z}_{12}$. I want to find the ...
user35360
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201
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Rings that are $K_0$ of finite groups
Is there a simple characterisation of all rings which appear as $K_0$ of finite groups? By $K_0$ of a finite group $G$ I mean $K_0(\mathbb C[G])$ which in the same as a ring of virtual characters of ...
- 2,147
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175
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Fusion pattern in a cyclic subgroup of order 8
Can a finite simple group $G$ have an element $x$ of order 8 such that $x^2$ is conjugate to $x^{-2}$ but $x$ is not conjugate to any element of $\{x^3,x^5,x^7\}$?
In other words, does this fusion ...
- 10.5k
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482
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Intersections of anisotropic tori with split Levi subgroups
Let G be a connected reductive group defined and split over a finite field k with Frobenius morphism F. Let T be an F-stable minisotropic maximal torus. Let P be an F-stable proper parabolic ...
- 211
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171
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Finding a database of representations as matrices
Sorry if this would be more appropriate as a stackoverflow and not a mathoverflow question, but I think it's more likely to be known in this community.
There are plenty of places on the internet or ...
- 7,902
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428
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The maximal order of an element in orthogonal groups over finite fields of characteristic 2
Let $q$ be a power of $2$ and let $(V,Q)$ be a
quadratic space of dimension $2m$ over $\mathbb{F}_q$. Up to isometry, we know that we have exactly two classes of such quadratic spaces: the plus type ...
- 7,521
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548
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When is Hom(G, H) the same size as Hom(H, G)?
Let $G$ and $H$ be finite groups. Consider the ratio
$$r_{G, H} \equiv {|Hom(G, H)| \over{|Hom(H,G)|}}$$
My question is
When is $r_{G, H} = 1$? Can we characterize the pairs of groups $(G, H)$ ...
- 71
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answers
115
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Sylow subgroups of the restricted Burnside group $\mathrm{RB}(d,n)$?
$\DeclareMathOperator\RB{RB}$What is known about the Sylow subgroups of the restricted Burnside groups $\RB(d,n)$ ?
I am looking for a reference.
In fact my question is slightly more general. Recall ...
- 61
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189
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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 ...
- 19.5k
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0
answers
116
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What are Burnside's "fixed systems" in modern language?
I just read Chapter 17, on rational invariants, in Burnside's classic 1911 text Theory of Groups of Finite Order. It was a great read, and mostly it was straightforward to translate the ideas into ...
- 2,831
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315
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(CFSG-free) Finite simple groups whose character degrees square divide its order
Let $G$ be a finite group. It is well-known that for all irreducible complex character $\chi$ then $\deg(\chi)$ divides $\lvert G\rvert$. Motivated by some problems with modular tensor categories, we ...
- 25.9k
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0
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121
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Finite linear groups with integer entries
Is there a classification of maximal finite subgroups of $\text{GL}_n(\mathbb{Z})$
for fixed $n$?
For $n$ sufficiently large the signed symmetric group is the subgroup of maximal order,
but is ...
- 83
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0
answers
150
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Subalgebra of group algebra generated by idempotents
Let $G$ be a finite group, and let $A$ and $B$ be two abelian subgroups of $G$. Let $K$ be a number field such that all characters of $A$ and of $B$ take values in $K$. Let $\mathcal{O}_K$ be the ring ...
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