Questions on group theory which concern finite groups.

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What are the possible automorphism groups of Riemann surfaces of low genus?

Skipping the easy cases of genus 0 and 1, what groups can arise as the group of conformal transformations of a Riemann surfaces of genus, say, 2 or 3? I'm frustrated because there are papers that ...
14
votes
1answer
696 views

The number of group elements whose squares lie in a given subgroup

This number is divisible by the order of the subgroup http://arxiv.org/abs/1205.2824. The proof is short but non-trivial. Is this fact new or is it known for a long time?
14
votes
2answers
584 views

Maximal number of maximal subgroups

Let $G$ be a finite group. I want to find an upper bound on the number of the maximal subgroups. My questions is does it possible to prove that the number of maximal subgroups of any finite group $G$ ...
14
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2answers
1k views

Groups with all normal subgroups characteristic

Today in my research, I had to use fairly explicitly the rather tautological property of finite cyclic groups that every normal subgroup is characteristic, i.e. fixed by all automorphisms. This got me ...
14
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2answers
455 views

factorization of the regular representation of the symmetric group

Let $\mathbb{C}[S_n]$ be the regular representation of the symmetric group $S_n$, and let $\mathbb{C}^n$ be the vector representation. Question: Does there exist a representation $V$ (of dimension ...
14
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1answer
466 views

Does O'Nan-Scott depend on CFSG?

My question is in the title. Some context: there are two versions of the O'Nan-Scott theorem. The first, weaker version, is due to O'Nan and Scott (independently) and gives the structure of the ...
14
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2answers
944 views

Number of n-th roots of elements in a finite group and higher Frobenius-Schur indicators

This is the second follow-up to this question on square roots of elements in symmetric groups and is concerned with generalisations to $n$-th roots. Let $G$ be a finite group and let $r_n(g)$ be the ...
14
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1answer
1k views

M24 moonshine for K3

There are recent papers suggesting that the elliptic genus of K3 exhibits moonshine for the Mathieu group $M_{24}$ (http://arXiv.org/pdf/1004.0956). Does anyone know of constructions of $M_{24}$ ...
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0answers
574 views

What's the big deal about $M_{13}$?

$M_{13}$ is the Mathieu groupoid defined by Conway in Conway, J. H. $M_{13}$. Surveys in combinatorics, 1997 (London), 1–11, London Math. Soc. Lecture Note Ser., 241, Cambridge Univ. Press, ...
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5answers
537 views

Results about the order of a group forcing a particular property.

Given a group of order $n$ where $n$ is either a specific number, or a number of a particular form, e.g. square-free, when does $n$ completely determine a particular group property among all groups of ...
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3answers
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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 ...
13
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3answers
690 views

Reference for representation theory of SL_2(Z/n)

There are many references for the representation theory (say over $\mathbf C$) of $SL_2(\mathbf{F}_q)$ and $GL_2(\mathbf{F}_q)$, for instance lecture 5 in Fulton--Harris "Representation theory" and ...
13
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4answers
2k views

determinant of the table of characters

I am certain that the answer to this question exists somewhere. It might be a classical exercise. Let $G$ be a finite group. Its table of characters is a square matrix, whose rows are indexed by the ...
13
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3answers
958 views

Restriction from $GL_n$ to $S_n$

Let $V$ be the irreducible representation of $GL_n$ with highest $\lambda$, and $|\lambda|=n$. It is well known that the representation of $S_n$ on the $(1,1,\ldots,1)$ weight space is the Specht ...
13
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3answers
406 views

Explicit construction of an element of ${\rm GL}(2, p)$ of order $p+1$

It is well-known that the order of $GL(2, p)$ is $(p^2-1)(p^2-p) = (p-1)^2(p+1)p.$ It is easy to construct matrices of orders $(p-1)$ and $p$ (diagonal and parabolic, respectively), but the only way ...
13
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1answer
455 views

Why is Klein's representation of $PSL_2(\mathbb{F}_7)$ hard to obtain?

In his famous article [1] Klein constructs a representation of $G=PSL_2(\mathbb{F}_7)$ in $\mathbb{C}^3$ (of which the first invariant polynomial of three variables gives rise to the famous Klein's ...
13
votes
2answers
497 views

Origin of group theory problem (bound on number of Sylow subgroups)

This problem (prove that the number of Sylow subgroups of a finite group $G$ is bounded by $\frac{2}{3}|G|$) posted on MSE proved rather difficult to solve. The OP has been silent about where the ...
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3answers
775 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
580 views

The prime divisors of a simple group

Let $G$ be a finite simple group such that $\pi (G)=\pi (A_{n})$, where $n\geq 23$ ($\pi (G)$ is the set of prime divisors of the order of $G$). Is it true that $G$ is isomorphic to an alternating ...
13
votes
1answer
487 views

Enumeration of $0-1$ matrices with determinant $1$

Has the number $f(n)$ of $n \times n$, $0{-}1$ matrices whose determinant is $+1$ been enumerated? E.g., for $n{=}2$, there are $f(2)=3$ such matrices: $$ \left( \begin{array}{cc} 1 & 0 \\ 0 ...
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0answers
632 views

How much has been written down about Deligne's geometric approach to the order formula for a finite group of Lie type?

This is a follow-up to a recent mathoverflow question 34387 about computing the orders of finite unitary groups and the comments made there. Between 1955 (Chevalley's Tohoku paper) and 1968 ...
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3answers
680 views

Non-commutator in simple group?

Hi, For a group $G$, we say that $x\in G$ is a commutator if there exists $a,b \in G$ such that $x=a^{1}b^{-1}ab$, and we say that $x$ is a non-commutator if there is no $a,b \in G$ such that ...
12
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4answers
908 views

Number of squares in a finite group

This was asked at MSE but never answered. Let $G$ be a finite group and denote by $sq(G)$ the number of squares in $G$ i.e. the number of elements in $G$ which possess a square root. For example, if ...
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4answers
1k views

Finite subgroups of $PGL_2(K)$ in characteristic $p$

Let $K$ be a field of characteristic $p$. What are the finite subgroups of $PGL_2(K)$ whose orders are divisible by $p$? And if $G$ and $H$ are two such subgroups that are isomorphic, can one say when ...
12
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5answers
2k views

Cosets and conjugacy classes

I'm interested in the following situation: $G$ is a finite group; $C$ is a conjugacy class in $G$; $H$ is the centralizer of an element $h$ of $C$. I want information on $|C\cap Hg|$ as $g$ varies ...
12
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4answers
1k views

Finite groups in which every character has real values: grading the representations

Let $G$ be a finite group. Then the irreducible complex representations of $G$ come in three sorts: real, complex and symplectic=quaternionic. The type of an irreducible character $\chi$ can be read ...
12
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2answers
605 views

Is this a well-known bound for the derived length of a finite group?

Let $cd(G)$ be the set of degrees of the irreducible complex characters of the finite group $G$. It is conjectured that if $G$ is solvable then $dl(G)\leq |cd(G)|$ and it is a result by Gluck that ...
12
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5answers
1k views

What is the Hilbert class field of a cyclotomic field?

In the answers to Qiaochu's post on defining representations of finite groups over the algebraic integers, it came out that which fields a representation of a finite group is defined over might depend ...
12
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3answers
2k views

How many conjugacy classes of subgroups does GL(2,p) have?

How many conjugacy classes of subgroups does $\mathrm{GL}(2,p)$ have? For instance the dihedral group of order $2n$ has $\tau(n)$ cyclic normal subgroups and $\sigma(n)$ "dihedral" subgroups (as ...
12
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2answers
690 views

Wedderburn's theorem for $\mathbb{Q}G$

Let $G$ be a finite group and let $\mathbb{Q}G=M_{n_1}(D_1)\times\cdots\times M_{n_k}(D_k)$ be the decomposition of $\mathbb{Q}G$ as a product of rings of matrices over divisions rings. Let $Z_i$ be ...
12
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1answer
258 views

Largest subset of $GL_n(p)$ in which pairwise subtraction is also in $GL_n(p)$

Suppose $X\subset \mathrm{GL}_n(p)$ is a set of invertible matrices such that for every $A,B\in X$ then also $A-B\in \mathrm{GL}_n(p)\cup \{0\}$. (If anyone knows a name for such sets I would be ...
12
votes
1answer
752 views

Find finite groups $G\cong Aut(G)$

I become interested in this problem because $G\cong Aut(G)$ suggests a special symmetry in $G$ (This kind of group describes its own symmetry). From $G/Z(G)\cong Inn(G)$ we know complete group is the ...
12
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3answers
911 views

How to find more (finite almost simple) groups with a given Sylow subgroup

I'm looking for some examples of actions on Sylow p-groups, and often those actions appear in the case of finite almost simple groups. Given a finite almost simple group, I understand in principle ...
12
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1answer
356 views

Crossed modules and degree-3 group cohomology

It is well known (see e.g. K. Brown, "Cohomology of groups") that a degree-3 cohomology class of a group G with coefficients in a module A can be thought of as an equivalence class of crossed modules, ...
12
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2answers
653 views

Is the following map from Z(G) x H^3(G, C*) --> H^2(G, C*) ever nontrivial?

Suppose that G is a finite group, then we have the following map f which takes an element z in the center of G and a 3-cohomology class w and returns a 2-cohomology class f(z,w) (for concreteness ...
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1answer
277 views

Variety of nilpotent Lie algebras or $p$-groups

Here's a couple of analogous questions, one in terms of finite-dimensional complex Lie algebras and one in terms of finite $p$-groups; I'd be interested in an answer to either: 1) Let $\mathcal{L}$ ...
12
votes
1answer
90 views

An inequality for the minimal number of generators of a finite group

Let $G$ be a finite group, $n(G)$ the minimal number of generators and $m(G)$ the minimal number of irreducible complex representations generating (with $\otimes$ and $\oplus$) the left regular ...
12
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1answer
289 views

Group cochains invariant under the action of the symmetric group

Let $G$ be a finite group and $A$ an abelian group. Recall the cochain groups $$ C^k = \{f: G^k \to A\} $$ and the coboundary map $$ \delta : C^k \to C^{k+1} $$ $$ (\delta f)(g_1, \ldots, g_{k+1}) ...
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0answers
669 views

What can be said about Schur indices, given only the character table?

Let $\chi$ be an irreducible (complex) character of a finite group, $G$. The Schur index $m_{K}(\chi)$ of $\chi$ over the field $K$ is the smallest positive integer $m$ such that $m\chi$ is afforded ...
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votes
3answers
1k views

Estimate for the order of the outer automorphism group of a finite simple group

It is known (given CFSG) that all non-abelian finite simple groups have small outer automorphism groups. However, it's quite tedious to list all the possibilities. Does anyone know a reference for a ...
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4answers
2k views

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 ...
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4answers
773 views

Realizable Order Sequences for Finite Groups

My post is motivated at least in part by this MO question. Has there been any work done on realizable order sequences for finite groups? By an "order sequence" I mean a non-decreasing list of the ...
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3answers
711 views

Dimensions and number of complex irreducible representations for SL3(Z/pZ)

This is my first post here :) I have the following two related questions. While looking in Conway's ATLAS (the 1985 one) for $SL_3(Z/pZ)$, where he does the cases $p=2,3,5,7$, I saw that the ...
11
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2answers
866 views

The maximum order of finite subgroups in $GL(n,Q)$

For several years people have tried to characterize finite groups of maximal order in $\mathrm{GL}(n,\mathbb{Q})$ (or $\mathrm{GL}(n,\mathbb{Z})$) and their orders. It appear in many articles a ...
11
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1answer
412 views

Group theory conjecture on hurwitz groups

Conjecture: Let $p$ be a prime. Then the group $G := \langle a, b \ | \ a^2, b^3, (ab)^7, [a,b]^9, (([a,b]^4)b)^{2p} \rangle$ has a composition series of the form ${\rm PSL}(2,8) - {\rm Z}_p - ...
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3answers
1k views

Number of generators of a subgroup of a finite simple group

For a finite group $G$ we denote $d(G)$ the minimal size of a set of generators of $G$. We define $D(G) = \max( d(H) \mid H\leq G)$. Let $S$ be a finite simple group. Are there `good' bounds on ...
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3answers
1k views

Is there a purely group-theoretic reformulation of an equivalence of subgroups?

There is an equivalence relation between inclusion of finite groups coming from the world of subfactors: Definition: $(H_{1} \subset G_{1}) \sim(H_{2} \subset G_{2})$ if $(R^{G_{1}} \subset ...
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1answer
402 views

Is every finite group a proper quotient of a finite primitive group?

Let $G$ be a finite group. Is there necessarily a finite primitive permutation group $P$ and a normal subgroup $N>1$ of $P$ such that $P/N \cong G$? If not, what restrictions are there on ...
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1answer
1k views

Extension of induced reps over Z: is it a sum of induced reps?

Let $G$ be a finite group. If $L$ is a finite free $\mathbf{Z}$-module with an action of $G$, say $L$ is induced if it's isomorphic as a $G$-module to $Ind_H^G(\mathbf{Z})$ with $H$ a subgroup of $G$ ...
11
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1answer
483 views

On the order of finite simple groups

About the order of finite simple groups there exists a very interesting result which stated as follows: Let $G$ be an non-solvable simple group of order $g$. If $p\mid g$, where $p>g^{1\over 3}$ ...