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2answers
859 views

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 ...
8
votes
1answer
264 views

Obstruction to extension of non-abelian groups - finite example?

Let $G$ be a non-abelian group, let $\Pi$ be a group, and let $\eta: \Pi\rightarrow Out(G)$ be a homomorphism, where $Out(G)$ is the group of automorphisms of G modulo the normal subgroup of inner ...
8
votes
1answer
396 views

When Cayley graphs of the symmetric group wrt generating sets of transpositions are isomorphic?

Dear All, I thought the following question might be well-known, but couldn't find anywhere, so decided to ask here: Let $A$ and $B$ be two generating sets for $S_n$, consisting of transpositions. ...
8
votes
1answer
364 views

Radical of $F_p[SL(2,p)]$

Let $G=SL(2,p)$. Does anyone know what is the radical of the group algebra $F_p[G]$? Does there exists any book/paper where it is calculated? By radical here I mean maximal ideal I of $F_p[G]$ such ...
8
votes
1answer
258 views

Characterization of Frobenius complements

I have learned that Frobenius complements are characterized (among finite groups) by having a fixed point free complex representation. That is, a finite group $G$ is a Frobenius complement if and only ...
8
votes
2answers
916 views

Generalization of a theorem of Øystein Ore in group theory

Theorem (Øystein Ore, 1938): A finite group $G$ is cyclic iff its lattice of subgroups $\mathcal{L}(G)$ is distributive. Proof: see below. Let $(H \subset G)$ be an inclusion of finite groups and ...
8
votes
2answers
264 views

3-cocycle representatives for the dihedral group $D_{2n}$?

I am looking for a reference for a complete list of 3-cocycle representatives for $H^3(D_{2n},\mathbb{C}^\times)$, where $$ D_{2n}=\langle a, b\mid a^2=b^2=(ab)^n=e\rangle $$ is the dihedral group of ...
8
votes
2answers
360 views

Parker-like loop of order 2187?

The Parker loop has order $2^{13}$, and reducing it modulo its 'center' yields -- not just a $12$-dimensional $\mathbb{Z} /(2)$-vector space, but one which can be naturally identified with the ...
8
votes
1answer
447 views

Groups with an automorphism of order two fixing only two elements

It is well known that a finite group admitting an automorphism of order 2 that fixes only the identity is abelian and has odd order. Moreover, the automorphism is inversion. Is anything known about ...
8
votes
1answer
295 views

Reference for restriction of a simple module over a splitting field to a smaller field?

This is mainly a request for a straightforward reference (preferably at textbook level). The question comes up while responding to a question raised by non-specialists in finite group ...
8
votes
1answer
463 views

Cohomology of orthogonal and symplectic groups

Hello, in their book Cohomology of Finite Groups Adem and Milgram investigate the cohomology of the finite orthogonal and symplectic groups only in case $\mathbb{F}_2$. Let $p$ be a prime dividing ...
8
votes
2answers
1k views

Number of faithful representations of a finite group

Is it known how many faithful linear representations a finite group G has on a complex vector space of given dimension? What if G is abelian? I would even be interested in this special case: the ...
8
votes
3answers
536 views

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 ...
8
votes
1answer
250 views

Condition for a certain subset being a subgroup

For any finite group $G$ and $n$ a divisor of $|G|$, consider the following subset of elements of "co-order" dividing $n$: $$G(n) = \{ g \in G \mid g^{|G|/n} = 1 \}$$ By a classical theorem of ...
8
votes
1answer
406 views

Homomorphic images of a Cartesian product of finite groups

What can be said about the class of groups which can be represented as a homomorphic image of an (infinite) Cartesian product (=unrestricted direct product) of finite groups? What would be simple ...
8
votes
1answer
432 views

Bounding the number of character degrees of a finite group in terms of the order of the group

Let $cd(G)$ be the set of degrees of irreducible complex characters of the finite group $G$ (so $cd(G) = \{\chi(1) | \chi\in Irr(G)\}$). What bounds are known of the form $|cd(G)|\leq f(|G|)$ (ie, ...
8
votes
1answer
488 views

Can one find the size of a Sylow normalizer from the character table?

Is the size of the normalizer of a Sylow p-subgroup determined by the ordinary character table of the group? And if so, how does one calculate it? In a solvable group, apparently one can compute ...
8
votes
2answers
298 views

Can a positive measure subset of a free group be nowhere dense?

Let $F$ be a finitely generated free group and let $S \subseteq F$ be a subset for which there is some $\epsilon > 0$ such that for any epimorphism to a finite group $\phi \colon F \to G$ we have ...
8
votes
0answers
274 views

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, ...
7
votes
3answers
359 views

Subgroups of $GL(k,q)$ for bounded $k$

This question on subgroups of $GL(2,q)$ asked by Jan, and especially wonderful answers to it given by Geoff Robinson, Ralph, and Will Sawin showing that "almost no finite groups" inject in $GL(2,q)$ ...
7
votes
4answers
380 views

Normal Covering of a Finite Group

Suppose $G$ is a finite group and $N_1, N_2, \cdots, N_k$ are proper normal subgroups of $G$. The set $\{ N_1, \cdots, N_k\}$ is called a normal cover for $G$, if $G = \cup_{i=1}^kN_i$. I need to the ...
7
votes
2answers
964 views

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?
7
votes
3answers
2k views

Faithful characters of finite groups

Related to a previous question I am asking furthermore a proof for the following: Question 1: If $\chi$ is a faithful irreducible character of a finite group $G$ then the regular character of $G$ is ...
7
votes
4answers
1k views

Presentation of the Monster Group

I was reading about the monster group, and how hard it was to do calculations in it, and I wondered: Is there a known presentation of the monster group? I know that it is a hurwitz group, but other ...
7
votes
5answers
624 views

Reference requested: Random walk on groups

I am looking for a good reference to learn about random walks on groups (either finite groups or Lie groups). Ideally, I would like a reference for general theory of random walks on groups that is ...
7
votes
3answers
605 views

Is there a Cayley graph of a non-abelian group that is not isomorphic to any Cayley graph of any abelian group?

It's the first question I post here :) I'm sorry if the question is too specific or if it's somehow repeating others. In other words, my question is the following. Consider a Cayley graph $\Gamma$ of ...
7
votes
2answers
421 views

What are the outer automorphisms of a Coxeter group?

I want to know the outer automorphisms of the Weyl group of $\mathrm{E}_8$, if any. But I might as well ask the question more generally. Suppose we have a Coxeter diagram. This gives a Coxeter ...
7
votes
2answers
285 views

Subgroup property stronger than being characteristic

In what follows, all groups are assumed to be finite. Recall that if $K \leq H \leq G$ are groups, $K$ is said to be a weakly closed subgroup of $H$ in $G$ if, for all $g \in G$, $g^{-1}Kg \leq H$ ...
7
votes
2answers
927 views

How to compute all irreducible representations of a finite group ? (how GAP is doing this?)

Let us "take" a finite group G. Here "take" I mean any type of group-theoretic description you prefer: e.g. as an explicit subset of GL (or other group) or Cayley table, whatever. Question: How ...
7
votes
2answers
620 views

characters on a finite group with `extremal' behaviour

The following question is a bit technical, and I haven't got to grips with it enough to be able to present it as a well-focused question. However, my hope is that the collective group-theoretic ...
7
votes
2answers
386 views

Finite groups with centerless quotients

Is there a description of finite groups whose all quotients have trivial center? Is it true that only direct products of non-abelian simple groups have this property?
7
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2answers
429 views

Character table entries and sums of roots of unity

It is well-known that the entries of the character table of a finite group are sums of roots of unity. Question: Is the converse true? Explicitly, given $z\in \mathbb{Z}[\mu_\infty]$, can I find a ...
7
votes
4answers
720 views

Is the classification of finite p-groups a smooth problem?

Fix a prime $p$. Then the question is about the difficulty of classifying finite $p$-groups. Originally I was going to ask if this was a "wild" problem but thanks to Joel David Hamkins' answer to ...
7
votes
1answer
366 views

Paths in groups

Given a finite group $G$, write $K(G)$ for the complete digraph on the elements of $G$. Label the edge from $g$ to $h$ by element $g^{-1}h$. Question: For what groups does there exist a Hamiltonian ...
7
votes
4answers
2k views

Orthogonal Groups over finite fields

Hello Let $\mathbb{F}_p$ be the finite field with $p$ elements. One can show that over finite fields, there are just two non-degenerate quadratic forms. So here I want to pick any non-degenerate ...
7
votes
1answer
410 views

Is there a precise notion of “almost all” such that almost all finite groups are Galois groups of extensions of the rationals?

I vainly tried to define a notion of "almost solvable group" such that every "almost solvable group" is the Galois group of a finite extension of the rationals, but I can't figure out the right way to ...
7
votes
2answers
440 views

Automorphism Group of a p-group : Looking for a Reference

In the following post by DavidLHarden : See Here He quoted the following claim: "There is a theorem that says that if $p$ is a prime and $|G|=p^n $ , then $|AutG| $ divides $ \Pi_{k=0}^{n-1} ...
7
votes
2answers
532 views

What groups have a second maximal subgroup below exactly four maximal subgroups?

I am looking for a finite group $G$ with the following property: there is a (core-free) subgroup $H < G$ such that the interval $\{ K : H < K < G\}$ in Sub[$G$] contains exactly four maximal ...
7
votes
2answers
908 views

classification of small complete groups

This is "escalated" from stackexchange. So, $S_n$ for $n \ne 2,6$, $\text{Aut}(G)$ for $G$ non-cyclic simple, $\text{Hol}(C_p)$ for $p$ odd prime are well-known classes of complete group. What's ...
7
votes
3answers
1k views

Steinberg Representations of Finite Groups of Lie Type

Let G be a finite group of Lie type. Assume G is also of universal type. Is the Steinberg representation of G generic, i.e., does the Steinberg representation admit a Whittaker model? A Whittaker ...
7
votes
4answers
895 views

Center and representations of finite group - how are related ?

If finite group G has a center how does it influence the representations of this group ? And vice versa - can we see somehow the center (or some of its properties) from representations (from ...
7
votes
2answers
369 views

How many finite loops?

How many finite loops of order $n$ are there? I am interested in the exact values ​​of $n$ if $n <40$ or even reasonable estimates. I am also interested in formulae or bounds for all $n$. Note ...
7
votes
2answers
480 views

Good effective versions of theorems of Artin and Brauer

The theorem of Artin and Brauer of the title are the famous theorem in the theory of representation of finite groups. For example, Artin's theorem is the statement that for every character $\chi$ of ...
7
votes
1answer
438 views

Many p,q-Sylow subgroups

It is a fact that the symmetric groups have as many 2-Sylow subgroups as possible. More precisely, for all $n \geq 1$, the number of 2-Sylow subgroups in $S_n$ is exactly $n!/2^{\nu_2(n!)}$, which is ...
7
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2answers
302 views

Are there any abelian 2 groups with Complete Holomorphs Other than $C^2_2$ and $C^4_2$?

The title says it. I'm reviewing some group theory concepts I haven't touched in quite a while, since I don't teach abstract algebra in my current position, and could not find the answer to this ...
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2answers
774 views

Are vertex and edge-transitive graphs determined by their spectrum?

A graph is called vertex and edge transitive if the automorphism group is transitive on both vertices and edges. The spectrum of a graph is the collection (with multiplicities) of eigenvalues of the ...
7
votes
1answer
1k views

Burnside's Lemma and Geometry

I think one of the most interesting results in Elementary Group Theory is the so-called "Burnside's Lemma", counting the numbers of orbits of a (finite) group action. I wonder if there is any ...
7
votes
1answer
2k views

Sylow subgroups invariant under an automorphism

Let $G$ be a finite group and $\sigma$ an automorphism of $G$. Suppose $p$ is a prime and $\sigma$ has prime order $q \neq p$. It's easy to see that $\sigma$ fixes a Sylow $p$-subgroup of $G$ if ...
7
votes
1answer
420 views

Extending group actions on varieties

Let $X$ be a (irreducible) variety (over $\mathbb{C}$ if necessary, smooth orbifold if necessary), and $U\subset X$ a nonempty open subset, and let $G$ be a finite group with an algebraic action on ...
7
votes
1answer
152 views

Which finite p-groups occur as commutators of finite p-groups?

Let $p$ be a prime number. For which finite $p$-groups $H$ there is a finite $p$-group $G$ such that $[G,G] \cong H$?