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12
votes
1answer
252 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
2answers
626 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 ...
12
votes
0answers
547 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 ...
11
votes
3answers
594 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 ...
11
votes
3answers
913 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 ...
11
votes
4answers
723 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 ...
11
votes
3answers
572 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
votes
2answers
555 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 ...
11
votes
1answer
364 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 - ...
11
votes
2answers
809 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
votes
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 ...
11
votes
1answer
368 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 ...
11
votes
1answer
536 views

In what sense is the classification of all finite groups “impossible”?

I think there is a general belief that the classification of all finite groups is "impossible". I would like to know if this claim can be made more precise in any way. For instance, if there is a ...
11
votes
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
votes
1answer
488 views

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

I become interested in this problem because $G\cong Aut(G)$ suggests a special symmetry in $G$. From $G/Z(G)\cong Inn(G)$ we know complete group is the anewer for the simplest case, though this class ...
11
votes
4answers
455 views

Iterated semi-direct products

Let $G$ be a finite group. Suppose that we can write $G= A \rtimes B$ and also $A = C \rtimes D$. Further suppose that C is normal in $G$ (not just in $A$). Then can we write $G = C \rtimes E$ where ...
11
votes
3answers
858 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 ...
11
votes
1answer
411 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 $p>g^{1\over 3}$ ...
11
votes
1answer
234 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, ...
11
votes
1answer
258 views

Finite groups with few double cosets with respect to abelian subgroup

The following question is motivated by the study of certain tensor categories, namely integral near-group categories. Let $G$ be a finite group and $H\subset G$ be a subgroup. Is it possible to give ...
11
votes
1answer
865 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 ...
11
votes
3answers
769 views

“Antipodal” maps on regular graphs?

This question is related to Realizing the diameter of a finite regular graph Let $X=(V,E)$ be a finite, connected, regular graph of diameter $D$. Assume that, for every vertex $x\in V$, there exists ...
11
votes
1answer
277 views

What are smallest finite images of triangle groups?

Given integers $a, b, c$ all at least 2, I would like to identify the smallest group with an element $x$ of order $a$, element $y$ of order $b$ such that the product $yx$ has order $c$. For example, ...
11
votes
1answer
692 views

Non-isomorphic two-transitive permutation groups with isomorphic point stabilizers

The permutation groups $A = PSL(2,7)$ with its natural action on the projective line $\mathbb{P}^1(\mathbb{F}_7)$ and $B = A\Gamma L(1,8)$ with its natural action on the affine line $\mathbb{F}_8$ ...
11
votes
2answers
556 views

(weak?) BN-Pair / Tits System for Sporadic Groups

The structure of finite simple groups of Lie type of arbitrary rank can be described well via BN-pairs. BN-pairs basically generalize the Bruhat decomposition of matrices into monomial $N$ and ...
11
votes
1answer
246 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}) ...
11
votes
0answers
254 views

Generating random finite groups

I would like a method to efficiently generate a random finite group of a given order $n$. If there are $g(n)$ non-isomorphic groups of order $n$, ideally each group would occur with probability ...
10
votes
3answers
920 views

Sylow subgroups of projective general linear groups

What is known about the Sylow 2-subgroups of $\rm{PGL}_n(\mathbb{F}_q)$, where $q$ is any prime power? For example, according to Theorem 7.9 in Isaacs's Character Theory, these Sylow 2-subgroups ...
10
votes
2answers
521 views

(co)homology of symmetric groups

Let $S_n=\{\text{bijections }[n]\to[n]\}$ be the n-th symmetric group. Its (co)homology will be understood with trivial action. What are the $\mathbb{Z}$-modules $H_k(S_n;\mathbb{Z})$? Using GAP, we ...
10
votes
4answers
313 views

The cohomology group $H^{1}(GL_{2}(\mathbb{F}_{p}), M_{2}(\mathbb{F}_{p}))$

Let $M_{2}(\mathbb{F}_{p})$ be the vector space of 2$\times$2 matrices over the finite field $\mathbb{F}_{p}$ where $p$ is a prime number, and let $GL_{2}(\mathbb{F}_{p})$ be the group of invertible ...
10
votes
4answers
726 views

Can we bound degrees of complex irreps in terms of the average conjugacy class size?

This question arises when looking at a certain constant associated to (a certain Banach algebra built out of) a given compact group, and specializing to the case of finite groups, in order to try and ...
10
votes
2answers
264 views

existence of a finite group which is the union of self normalizing subgroups

Can a finite group G be the union of self normalizing subgroups such that the intersection between any two of these subgroups is equal to the unit of the group G? I don't think so but I can't prove ...
10
votes
3answers
1k 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 ...
10
votes
3answers
582 views

Is the Gelfand-Graev character isomorphic to a cohomology group for some sheaf on a Deligne-Lusztig variety?

Deligne-Lusztig theory is awesome. You take a maximal torus $T$, you take a character $\theta$, construct a variety $X_T$$^*$, take etale cohomology, get a virtual character $R_T^\theta$, maybe it's ...
10
votes
1answer
451 views

Does every automorphism of G come from an inner automorphism of S_G?

I feel sort of silly asking this question. Unless I'm very much mistaken the paper I'm reading assumes the following statement: Let $G$ be a finite group. We may embed it via the Cayley embedding ...
10
votes
2answers
256 views

Which finite nonabelian groups have all their quaternionic representations of degree one?

A finite group $G$ has a finite set of irreducible representations over the complex numbers. All of these representations are linear (that is, are maps in 1x1 complex matrices) if and only if $G$ is ...
10
votes
3answers
943 views

Can we always find such an irreducible polynomial of degree n where degree(p(x)-x^n)<= n/2?

Consider unitary polynomials of degree $n$ over $GF(2)$. That is, polynomials of the form $p(x) = \sum_{i=0}^n a_i x^i$ where $a_i\in GF(2)$ and $a_n=1$. Can we always find such an irreducible ...
10
votes
3answers
855 views

(A very limited instance of) Lagrange's Theorem's converse and A_5

Suppose $G$ is a finite simple group and $|G|$ is a multiple of $60$. Does it follow that $G$ has a subgroup isomorphic to $A_{5}$? If so, can this be proven without using the Classification?
10
votes
2answers
559 views

Embeddings of finite groups into GL(n,Q_p)

This question is inspired by some interesting comments on this recent question. Fix an integer $n \geq 1$ and a finite subgroup $G$ of $\mathrm{GL}_n(\mathbf{C})$. It is known that there are ...
10
votes
3answers
843 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 ...
10
votes
4answers
701 views

Groups which satisfy Mal'cev's theorem (locally residually finite)

Recall that a group $G$ is residualy finite if for every non-zero element $g\in G$ there exists a homomorphism $\sigma:G\rightarrow H$ such that $H$ is finite and $\sigma(g)\neq 0$. Mal'cev's theorem ...
10
votes
3answers
487 views

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 ...
10
votes
4answers
757 views

Efficient presentations for finite groups

A finitely presented group which has more generators than relations has an infinite abelianization and so is an infinite group. Therefore, for a finite group, all presentations must have at least as ...
10
votes
2answers
443 views

Finite subgroups of $PGL(3,K)$

It is well-known that finite subgroups of $PGL_2(\mathbb{C})$ are cyclic groups, dihedral groups, A4, S4 and A5 and each of these groups occurs exactly once (up to conjugacy). These facts are ...
10
votes
2answers
354 views

Convenient reference for subgroups of a finite semidirect product?

Given a finite group $G= H \ltimes N$ (with no particular constraints on $H, N$), it's probably been known for a long time how to describe efficiently the possible subgroups of $G$. A graduate ...
10
votes
4answers
871 views

Diameter of universal cover

Let $M$ be Riemannian manifold and $\tilde M$ be its universal cover (with induced metric). What is the upper bound for $k=\mathop{diam}\tilde M/\mathop{diam} M$ in terms of $m=|\pi_1(M)|$ (or ...
10
votes
2answers
515 views

A group action of the Heisenberg group with special symmetries

Suppose we look at the Heisenberg group $H_{d}$ as a matrix group of upper triangular matrices over the ring $\mathbb{Z}/d\mathbb{Z}$. You can even choose $d$ to be prime if you want. A natural ...
10
votes
4answers
1k views

Mystery of the Monstrous Moonshine

There's a very famous group, the largest sporadic simple finite group, sometimes called a monster whose size is quoted below. What's the explanation that the primes appearing in it, ...
10
votes
1answer
449 views

complexity of counting homomorphisms

This is a question I have thought about and asked a number of people, but have never got an answer beyond "It should be true that..." Given a finitely generated group $G$ (eg. a link group ...
10
votes
0answers
276 views

$2$-group with two isomorphic normal subgroups of index $4$ with non-isomorphic quotients

Hypothesis: Let $P$ be a finite $2$-group with two isomorphic normal subgroups $M$ and $N$ such that $P/M\cong C_4$ (the cyclic group of order $4$) while $P/N\cong C_2^2$. By the lattice theorem, ...