Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

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31
votes
1answer
539 views

Known and fixed gaps in the proof of the CFSG

As the "second-generation" proof of the Classification of Finite Simple Groups is being written up in the volumes by Gorenstein, Lyons, Aschbacher, Smith, Solomon, and others (see e.g. this ...
2
votes
1answer
96 views

Automorphism groups of simple groups of Lie type

$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\PGL{PGL}$In “Automorphisms of finite linear groups”, Steinberg proves that any automorphism of a simple group of Lie type (normal or twisted) is a ...
1
vote
0answers
80 views

Conway's quaternion notation +1/3[C_3×C_3]∙2^(2) represent C_3h of 4d Point group?

In John Conway and Derek Smith's On Quaternions and Octonions: their Geometry, Arithmetic, and Symmetry, they introduce a way to connect quaternions to 4D Point Group. Suppose: $[l,r]:x\to \bar lxr\;,\...
49
votes
2answers
4k views

Where are the second- (and third-)generation proofs of the classification of finite simple groups up to?

According the the Wikipedia page, the second generation proof is up to at least nine volumes: six by Gorenstein, Lyons and Solomon dated 1994-2005, two covering the quasithin business by Aschbacher ...
2
votes
0answers
125 views

Can the relation between count of commuting pairs and conjugacy classes for finite groups be generalized to semigroups?

It is well-known that number of pairs of commuting elements in finite group G is equal to number of conjugacy classes multiplied by cardinality of G. More generally here (MO275769) Qiaochu Yuan ...
6
votes
1answer
263 views

On the density of the orders excluded by the Sylow theorems for simple groups

If $G$ is a finite group whose order is divisible by a prime $p$ and $p^r$ is the maximal power of $p$ that divides it, the Sylow theorems tell us that the number $n_p$ of Sylow $p$-subgroups of $G$ ...
4
votes
0answers
132 views

Subgroups of $\operatorname{GL}(n,q)$ transitive on non-zero vectors

Is there a classification of subgroups $G$ of $\operatorname{GL}(n,q)$ which act transitively on $\mathbb{F}_q^n \setminus \{0\}$, the set of non-zero vectors? Any $G$ with $\operatorname{GL}(n/m,q^m) ...
7
votes
0answers
175 views

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 ...
0
votes
0answers
37 views

How can I find the order of the elements of the maximal subgroups for G_2(3)?

I'm looking to find the maximal subgroups for the exceptional group of Lie type $G_{2}(3)$ using GAP. Currently I can do the following: ...
0
votes
0answers
35 views

Counting the number of generating triples of various types in finite simple groups

I am trying to figure out how specific generating triples in finite simple groups are calculated. My understanding is that it uses Frobenius's formula and character theory. I'm not an expert on ...
7
votes
0answers
89 views

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 ...
7
votes
0answers
170 views

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;...
-2
votes
0answers
55 views

Classification of minimal 2-groups with special property

I would greatly appreciate it if you kindly give me some advice to tackle the below situation. Let $G$ be the non-abelian 2-group $$G=\langle a, b, c \mid a^{2^n}=b^{2^m}=c^2=1, [a,b]=c, [a,c]=1, [b,c]...
3
votes
1answer
68 views

Bounds for the number of edges in an Alperin diagram

If $A$ is an algebra over a field $k$ and $M$ is a finite-dimensional $A$-module, then Alperin showed in a paper [Diagrams for modules, JPAA, 1980] how to associate a diagram to $M$ with the vertices ...
2
votes
1answer
109 views

How do I find hyperbolic generating triples for a group using GAP?

Let $G$ be a finite group and $x, y, z \in G$. A hyperbolic generating triple for $G$ is a triple $(x, y, z) \in G\times G\times G$ such that $\frac{1}{o(x)}+\frac{1}{o(y)}+\frac{1}{o(z)} <1$, $\...
4
votes
3answers
426 views

What are the rank 3 boolean intervals [H,G], with G simple group?

The rank $n$ boolean lattice $B_{n}$ is the subset lattice of $\{1,2, \dots , n\}$. The lattice $B_{3}$ is the following: Question: What are the rank $3$ boolean intervals of the form $[H,G]$, with $...
25
votes
4answers
964 views

$\operatorname{PSL}(2,\mathbb{F}_p) $ does not embed in $\mathfrak{S}_p$ for $p>11$

A famous result of Galois, in his letter to Auguste Chevalier, is that for $p$ prime $>11$ the group $\operatorname{PSL}(2,\mathbb{F}_p) $ does not embed in the symmetric group $\mathfrak{S}_p$. ...
17
votes
3answers
3k views

Finite groups such that every irrep can be induced from trivial irrep of a subgroup ?

What are examples (general features) of the finite groups $G$, such that every irrep (irreducible representation) is contained (as constituent) in the representation induced from trivial ...
41
votes
8answers
9k views

The finite subgroups of SU(n)

This question is inspired by the recent question "The finite subgroups of SL(2,C)". While reading the answers there I remembered reading once that identifying the finite subgroups of SU(3) is still an ...
15
votes
2answers
571 views

The finite groups with a zero entry in each column of its character table (except the first one)

$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Aut{Aut}$Consider the class of finite groups $G$ having a zero entry in each column of its character table (except the first one), i.e. for all $g \...
8
votes
0answers
99 views

Tensor products of irreducible representations of $GL_{2}(\mathbb{F}_{q})$

Throughout the post $G = GL_{2}(\mathbb{F}_{q})$ where $q$ is a prime power with the prime not being 2. Let $V_{1}$ and $V_{2}$ be cuspidal representations of $G$ over $\mathbb{C}$. I can understand ...
6
votes
0answers
137 views

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 ...
9
votes
1answer
237 views

How small can the support of a nontrivial $\mathbb F_p$-cocycle on $C_p$ be?

Let $p$ be a prime, and let $\phi : C_p^n \to \mathbb F_p$ be an $\mathbb F_p$-valued $n$-cocycle on $C_p$ (the cyclic group of order $p$) which is not an $n$-coboundary, i.e. $\phi$ represents a ...
30
votes
3answers
3k views

Order of products of elements in symmetric groups

Let $n \in \mathbb{N}$. Is it true that for any $a, b, c \in \mathbb{N}$ satisfying $1 < a, b, c \leq n-2$ the symmetric group ${\rm S}_n$ has elements of order $a$ and $b$ whose product has order $...
12
votes
3answers
427 views

Small simplicial set models for BG

Let $F$ be a finite group. Is there a model for $BF$ as a simplicial set such that the number of nondegenerate $n$-simplices grows at most polynomially? For example the Bar construction has the ...
2
votes
0answers
60 views

The number of orbits of a two-point stabilizer of the symplectic group $Sp(2m,2)$

I am trying to figure out the number of orbits of a two-point stabilizer of the action of $Sp(2m,2)$ on its two orbits $\Omega^+$ and $\Omega^-$ as detailed in Dixon and Mortimer's "Permutation ...
8
votes
2answers
1k views

Maximal order of finite subgroups of $GL(n,Z)$

I am interested in the finite subgroups of $GL(n,Z)$ of maximal order. Except for the dimensions $n = 2,4,6,7,8,9,10$ they are -- up to conjugacy in $GL(n,Q)$ -- in each dimension the group of signed ...
1
vote
2answers
332 views

Why are finite simple groups useful? [duplicate]

The classification of finite simple groups has been called one of the great intellectual achievements of humanity, but I don't even know one single application of it. Even worse, I know a lot of ...
25
votes
1answer
2k views

Number of 2-dimensional irreducible representations of a finite group ?

Question: What is the number of two-dimensional irreducible representations of a finite group ? How it can be expressed in groups-theoretic terms ? (Number of 1-dimensional irreps is |G/[G,G]| ). ...
0
votes
1answer
150 views

Irreducible representations of finite p-groups

Let $G$ be a finite $p$-group. What are irreducible representations of $G$ over a field of characteristic $q$, such that $(p,q)=1$ ? Can we say something in general ? In particular, if there exists ...
3
votes
3answers
443 views

Large product-1-free sets in finite groups

$\DeclareMathOperator\SmallGroup{SmallGroup}$Definition. A subset $A$ of a group $G$ is called product-1-free if for any sequence of pairwise distinct elements $a_1,\dots,a_n$ of $A$ the product $a_1\...
0
votes
0answers
107 views

Another question concerning finite metacyclic groups

Given a non-split finite metacyclic group $H$, does there always exist a finite split metacyclic group $G$ with a normal cyclic subgroup $N$ of prime power order such that $H \cong G/N$? Based on my ...
2
votes
1answer
151 views

Structures of subgroups of a finite abelian p-group

$\newcommand\la{\langle}\newcommand\ra{\rangle}$Let $G=\mathbb{Z}/p^{i_1}\times\cdots\times\mathbb{Z}/p^{i_r}$ with $i_1\leq\ldots\leq i_r$ be a finite abelian $p$-group. Then there can be many ...
0
votes
1answer
92 views

Perfect Cayley graphs for abelian groups have $\frac{n}{\omega}$ disjoint maximal cliques

Let $G$ be a perfect/ weakly perfect Cayley graph on an abelian group with respect to a symmetric generating set. In addition let the clique number be $\omega$ which divides the order of graph $n$. ...
4
votes
1answer
178 views

The action of a subgroup of the torsion group of elliptic curves on integral points?

Let $E$ be an elliptic curve given in long Weierstraß form with all coefficients $a_1,a_2,a_3,a_4,a_6 \in \mathbb{Z}$. It is known that the rational points $E(\mathbb{Q})$ form a group which has a ...
4
votes
1answer
203 views

Finite simple groups with the same numbers of elements of orders p and q

Let $G$ be a nonabelian finite simple group, and let $p$ and $q$ be distinct prime divisors of the order of $G$. Is it true that the number of elements of $G$ of order $p$ never equals the number of ...
4
votes
1answer
258 views

A question about the possibilities of GAP

Let $R=\mathbb{Z}/1024\mathbb{Z}$ and $G=GL(3,R)$. Let $H$ be the subgroup of $G$ consisting of all matrices with determinant $1$ which are congruent to the identity matrix modulo the ideal $4R$. Let $...
10
votes
3answers
580 views

Realizability of a real representation using real cyclotomic coefficients

Let $G$ be a finite group and $\rho: G \rightarrow GL(d,\mathbb{C})$ an irreducible representation with Frobenius-Schur indicator $\frac{1}{|G|}\sum_{g\in G} \operatorname{tr} \rho(g^2) = 1$. Thus $\...
4
votes
1answer
181 views

A transitive action on a specific set

Let $G$ be a finite group and $\lambda\in G$, consider a set $$D^{p+*}_{G}(\lambda):=\{P\in S^{p+*}_G|P^{\lambda}=P=[P,\lambda]\}$$ where: $S^{p+*}_{G}$ denotes the set consisting of all non-trivial $...
2
votes
0answers
88 views

Action of diagonal automorphisms on the set of irreducible characters of $D_n(q)$

Let $S$ be $D_n(q)$ where $q$ is a prime power. We know that a diagonal automorphism $\phi_h$ of $S$ is of the form $g\mapsto hgh^{-1}$, where $h\in \hat H$ and $\hat H$ normalizes $S$. Note that $\...
7
votes
3answers
462 views

Product-one sets in non-commutative groups

A nonempty subset $D$ of a group $G$ is called $\bullet$ decomposable if $D\subseteq DD$, that is every element $x\in D$ is can be written as the product $x=yz$ of some elements $y,z\in D$; $\bullet$ ...
18
votes
3answers
1k 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 ...
3
votes
1answer
113 views

Subgroups and representations of finite groups of Lie type

Is there a usable bound for the minimal index of a proper subgroup in a finite simple group of Lie type in terms of its rank and the characteristic (or even cardinality) of its field of definition? ...
11
votes
1answer
319 views

Number of 1's in binary expansion of $a_n = \frac{2^{\varphi(3^n)}-1}{3^n}$

My question is about the Hamming Weight (or number of 1's in binary expansion) of $a_n = \frac{2^{\varphi(3^n)}-1}{3^n}$ A152007 For example, $a_3 = 9709 = (10110111101001)_2 $ has nine 1's in binary ...
5
votes
0answers
115 views

Finite simple groups of automorphisms of finite simple Lie algebras

I begin by briefly recalling some basic facts in order to pose my question in context. According to the classification, the finite simple groups are cyclic of prime order, are alternating on $n \geq 5$...
3
votes
1answer
229 views

Computing characters of $\alpha$-projective representations

Given a finite group $G$, a finite cyclic group $A$ (viewed as a subgroup of $\mathbb{C}^{\times}$, i.e. generated by a $|A|$-th root of unity), and a 2-cocycle $\alpha\in Z^{2}(G,A)$. Recall that an $...
8
votes
1answer
151 views

Is there always a simple module whose Green correspondent is a simple module under some conditions?

Let $G$ be a finite group and $KG$ its group algebra over some field $K$ with $\mathrm{char}\ K$ dividing the order of $G$. It's well-known that the Green correspondence is compatible with the Brauer ...
4
votes
1answer
205 views

Finite groups with a dihedral maximal subgroup

Suppose $G$ is a finite group with a dihedral maximal subgroup. Suppose that $G$ is not isomorphic to $\operatorname{PSL}(2,q)$ for some any prime-power $q$. Is $G$ always solvable?
5
votes
3answers
276 views

Generation of permutation groups by fixed elements subgroups

Suppose $(H,X)$ is a permutation group (with $H$ a group acting faithfully on the set $X$). Under what circumstances is $H$ generated by its subgroups $H_x$, where $H_x$ is the subgroup of $H$ fixing $...
4
votes
1answer
255 views

Is the fixed subring a symmetric algebra?

Let A be a finite dimensional symmetric k-algebra over some field k. The set of units of A is denoted by U(A). Suppose G is a cyclic group of prime order which acts via inner algebra automorphism on A,...

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