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Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

Questions on group theory which concern finite groups. In addition to this tag, it is recommended to use also the more general tag .

2267 questions
16
votes
1 answer

Ratio of number of subgroups to the order of a finite group

Let $\mathcal{G}$ be the set of finite groups and for $G \in \mathcal{G}$, let $S(G)$ be the set of subgroups of $G$. I am interested in the ratio $R(G)=|S(G)|/|G|$. It is easy to show that by picking $G$ appropriately, $R(G)$ can be made…
jwellens
  • 413
15
votes
1 answer

Is there anything significant about GAP's SmallGroup(512,2045)?

Here's the output of the GAP command "SmallGroupsInformation(512)" There are 10494213 groups of order 512. 1 is cyclic. 2 - 10 have rank 2 and p-class 3. 11 - 386 have rank 2 and p-class 4. 387 - 1698 have rank 2 and p-class 5. 1699 - 2008 have…
David Treumann
  • 4,849
15
votes
2 answers

The Weyl group of E8 versus $O_8^+(2)$

Right now Wikipedia says: The Weyl group of $\mathrm{E}_8$ is of order 696729600, and can be described as $\mathrm{O}^+_8(2)$. The second part feels wrong to me. $\mathrm{O}^+_8(2)$ is the group of linear transformations of an 8-dimensonal vector…
John Baez
  • 21,373
11
votes
2 answers

Aut(G) = $C_3$, G = ?

Is there a group G such that Aut(G) = $C_3$? What if we replace 3 with a prime number p?
Greg Gibson
  • 423
10
votes
2 answers

Applications of fusion systems

What are the applications of theory of fusion systems to finite group theory or representation theory of finite groups? More concretely, is there any important result in finite group theory or representation theory of finite groups whose prove uses…
zamanjan
  • 689
9
votes
1 answer

Nonabelian finite groups with "locally commuting" presentation

Let $G = \left\langle S | R \right\rangle$ be a finitely presented group where S is a set of generators and R is a set of relations. We say that the presentation is "locally commuting" if whenever two generators $a, b$ appear in a word in R, the…
Jalex Stark
  • 141
9
votes
1 answer

A question about representations of finite groups

Let $G$ be a finite group and let $V$ be an irreducible complex representation of $G$. Does there exist an element $g \in G$ which acts on $V$ with distinct eigenvalues? If true, can you provide a proof/reference, and if false, a counterexample?
Henri Darmon
  • 91
9
votes
7 answers

maximal subgroups of finite simple groups

Is it possible to determine the structure of maximal subgroups of finite simple groups?(Even if in special cases such as minimal simple groups, alternating groups,...)
sebastian
  • 457
8
votes
3 answers

A Perturbation problem for U(n)

Let G be a finite subgroup of U(n), the unitary group acts on $\mathbb{C}^n$. If there is a unit vector $x$ in $\mathbb{C}^n$ such that g(x) is almost orthogonal to x, for all $g\in G$ except the identity, can we perturb x so that g(x) is exactly…
Qingyun
  • 411
8
votes
3 answers

Is there order to the number of groups of different orders?

I was always struck by how uncharacteristically erratic the behavior of the following function is: $f: \mathbb{N} \rightarrow \mathbb{N}$ given by $f(n):=$ number of isomorphism classes of groups of order $n$. I blame this on the introduction of the…
James D. Taylor
  • 6,178
7
votes
3 answers

Solvable transitive groups of prime degree

Is the following true ? Every solvable transitive subgroup $G\subset\mathfrak{S}_p$ (the symmetric group on $p$ letters, where $p$ is a prime) contains a unique subgroup $C$ of order $p$ and is contained in the normaliser $N$ of $C$ in…
Chandan Singh Dalawat
  • 18,449
7
votes
1 answer

Isomorphic simple groups

It is known that $SL_{4}(\mathbb{F}_2)\cong A_8$. Obviously, this is equivalent to the existence of a subgroup of $Sl_4(\mathbb{F}_2)$ of index $8$. How to find such a subgroup?
user30504
  • 71
6
votes
1 answer

The best upper bound for the number of involutions in a finite non-abelian simple group

Let $G$ be a finite non-abelian simple group and $t$ is equal to the number of involutions of $G$. We know that $t<|G|/3$ or $3t+1 \leq |G|$. Is this the best upper bound for the number of involutions of $G$? and if not what is it (if there is any)?
Ahmadi
  • 123
5
votes
1 answer

A problem in Finite Group Theory

This is a problem I encountered in Martin Isaacs' 'Finite Group Theory'. It's located at the end of Chapter II which deals with subnormality, and the particular paragraph is concerned with a couple of not so well-known results which I quote for…
user13040
5
votes
0 answers

Number of elements in the spectrum of a finite simple group

For a finite group $G$ one defines the spectrum $\omega(G)$ as the set of element orders of $G$. The set $\omega(G)$ is uniquely determined by the subset $\mu(G)$ consisting of those elements of $\omega(G)$ that are maximal with respect to the…
Timm von Puttkamer
  • 555
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