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Reference for this theorem in representation theory?

Let $G$ be a finite group and $\chi$ be an irreducible character of $G$ (characteristic zero algebraically closed base field). If $H$ is the kernel of $\chi$ then the irreducible representations of ...
8
votes
1answer
976 views

Incompressible surfaces in an open subset of R^3

Let $U$ be a connected open subset of $\mathbb R^3$. Furthermore, we have: $\mathbb R^3\setminus U$ has exactly two connected components (thus by Alexander duality, $H_2(U;\mathbb Z)=\mathbb Z$). ...
8
votes
4answers
868 views

What does the typical non-solvable group look like?

According to a result of Higman and Sims (which I learned about in this paper of Poonen's) the typical p-group is 3-step nilpotent of a particular form. In particular the typical group is a 3-step ...
8
votes
2answers
450 views

Easier reference for material like Diaconis's “Group representations in probability and statistics”

I'm teaching a class on the representation theory of finite groups at the advanced undergrad level. One of the things I'd like to talk about, or possibly have a student do any independent project on ...
8
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2answers
385 views

A conjecture on solvablity of finite groups

Suppose $G$ is a finite group and $A$ an abelian subgroup. Suppose for some natural number $n\geq 2$, elements of $\gamma_n(G)$ have the form $[a, x]$ where $a\in A$ and $x\in G$. Then $G$ is ...
8
votes
2answers
661 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
3answers
867 views

Which finite groups are generated by n involutions?

One of the interesting problems in abstract polytope theory is to determine, for a given finite group, when that group is the automorphism group of a regular abstract polytope. This is equivalent to ...
8
votes
3answers
531 views

The powers of non-empty subset of a group that generate a subgroup

If G is a group and A and B to non-empty subsets of G, then by AB we mean the set consist of all product ab where a is in A and b is in B.(Standard definition) Similarly we can define X^m where X is a ...
8
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1answer
196 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
2answers
332 views

Decomposition of induced representations / Refinement of Mackey's criterion

There are already some questions with almost the same title, but they are more restrictive. Let $G$ be a finite group, $H$ a subgroup, $V$ an irreducible representation of $H$, and $W=Ind_H^G V$ the ...
8
votes
1answer
342 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
294 views

A subgroup intersects conjugacy class of every prime power order element

Let $G$ be a finite group and $H$ be a subgroup of $G$. Suppose that for any prime power order element $x$ of $G$, there exists some element $g$ in $G$ such that $x^g$ is contained in $H$. Does it ...
8
votes
2answers
344 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
391 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
285 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
398 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
1answer
358 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
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1answer
411 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
464 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
1answer
469 views

Periodic Automorphism Towers

In Scott's classic textbook on Group Theory, he asks: Suppose that $G$ is a finite group. Is the sequence of isomorphism types of the groups $Aut^{(n)}(G)$ for $n \in \mathbb{N}$ eventually periodic? ...
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2answers
518 views

Automorphisms of subgroup of hamming cube under distance constraint

Let $S$ be a subset of $\{0,1\}^n$ such that any two elements of $S$ are at least (Hamming) distance 5 apart. I'm looking for an upper bound on the size of the automorphism group of $S$. There's a ...
7
votes
3answers
346 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
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2answers
943 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
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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
3answers
493 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 ...
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4answers
829 views

Textbook source for finite group properties deducible from character table?

Various questions have been posted on MO (some answered, some not) involving the character table of a finite group $G$ over a splitting field such as $\mathbb{C}$ of characteristic 0. My basic ...
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votes
4answers
629 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
2answers
576 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
886 views

What about the classification of big finite simple groups?

How hard is it to classify all big finite simple groups, i.e., all finite simple groups larger than some sufficiently large constant? Alternatively - how hard is it to classify all finite simple ...
7
votes
2answers
348 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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4answers
692 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
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2answers
556 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
400 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
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2answers
519 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 ...
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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
2answers
188 views

On non-split extensions of $\mathrm{SL}_d(q)$

Throughout this question, the following notation holds: Let $q$ be a power of a prime $p$, and let $d>4$ be a positive integer. Let $G$ be a finite group with a normal subgroup $E$ which is an ...
7
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2answers
294 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 ...
7
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2answers
700 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
2answers
183 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 ...
7
votes
2answers
343 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
1answer
460 views

An extension of the converse to Hall's theorem.

This is an extension of this MSE question, in which I asked whether there was a counterexample to the following statement, Conjecture. If a finite group $G$ contains a $\lbrace p,q \rbrace$-Hall ...
7
votes
2answers
399 views

Regular elements in the torus of a group of Lie type

Let $G$ be a simple linear algebraic group, and $F$ a Frobenius map, i.e. some power of $F$ is the standard Frobenius map which raises matrix entries to the $q$-th power. Then $G^F$ is a group of Lie ...
7
votes
1answer
336 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. ...
7
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1answer
964 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 ...
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2answers
314 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
1k 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
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1answer
401 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
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1answer
353 views

Octonionic reflection groups

Consider Leech lattice definition provided by Wilson (Octonions and Leech lattice, 2008). There are 819 E8 sublattices defined by $ (2\lambda, 0, 0); $ $ (\lambda \overline{s}, (\lambda ...
7
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3answers
516 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 ...