Questions on group theory which concern finite groups.

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23
votes
2answers
1k views

Monstrous Moonshine for Thompson group $Th$?

I. As a background, in Traces of Singular Moduli (p.2), Zagier defines the modular form of weight 3/2, $$g(\tau) = \frac{\eta^2(\tau)}{\eta(2\tau)}\frac{E_4(4\tau)}{\eta^6(4\tau)}=\vartheta_4(\tau)\, ...
3
votes
4answers
2k views

Automorphisms of non-abelian groups of order $ p^3$

There are two non-abelian groups of order $p^3$, namely, semi-direct product of $\mathbb Z /p \mathbb Z \times \mathbb Z /p \mathbb Z$ by $\mathbb Z /p \mathbb Z$ and semi-direct product of $\mathbb Z ...
3
votes
1answer
373 views

On the Groups of Order $(p^2+1)/2$

A few days ago I asked a question (Groups of order $p(p^2+1)/2$) about a finite group of order $p(p^2+1)/2$ and I got a lot of useful information about it. Thanks for the nice and very helpful answers....
9
votes
4answers
661 views

Classification of automorphism groups of groups of order $p^4$

For the purpose of classifying another algebraic structure which is parametrised by the choice of a group and of an automorphism I need the classification up to isomorphism of automorphism groups of p-...
0
votes
0answers
78 views

a modular character problem [closed]

Let $B\in$Bl$(G|D)$ and suppose that $\sigma\in$Aut$(G)$ fixes every $\chi\in$Irr$(B)$. If $d\in D$, show that $d$ and $d^\sigma$ are $G$-conjugate. It is a problem from Navarro's book "characters and ...
11
votes
4answers
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 ...
4
votes
1answer
154 views

Sum of Young symmetrisers of a given shape

Preliminaries and notation: Let $n\in \mathbb{Z}_{>0}$ and $\lambda=(\lambda_1,\lambda_2,\dots,\lambda_s)\vdash n$ be a partition. Given a Young diagram of shape $\lambda$, we can associate it ...
-4
votes
0answers
97 views

Finite groups are isomorphic [closed]

For two finite groups $G_1, G_2$ if for every integer $n\geq 0$, $|G_1^n| = |G_2^n|$, then is it true that $G_1\cong G_2$? By $G^k$ we mean set $\{g^k|g\in G\}$.
3
votes
1answer
199 views

Must normalizing field outer automorphisms “divide” the dimension?

Imprecise question: To get a normalizing field outer automorphism of order $r$, must we multiply the dimension by $r$? Precise hypothesis: Let $p\geqslant 5$ be a prime, let $q$ be a power of $p$ and ...
1
vote
1answer
130 views

p-groups with special property on its centralizers

Thanks for any help or comment. Suppose $G$ is a finite non-abelian p-group. Suppose $G$ has a proper non-abelian subgroup $M$ such that for every non-central element $x\in M$, $C_G(x)\subseteq M$. ...
4
votes
3answers
227 views

Index of agemo subgroups in $p$-groups

Having a finite $p$-group $G$ ($p$ odd). we denote by $\Omega_1(G)$ the subgroup generated by all the elements of $G$ of order dividing $p$. Is there an example of such a group $G$, such that $|G:...
29
votes
3answers
1k views

Richness of the subgroup structure of p-groups

Given a prime $p$ and $n \in \mathbb{N}$, let $f_p(n)$ be the smallest number such that there is a group of order $p^{f_p(n)}$ which all groups of order $p^n$ embed into. What is the asymptotic growth ...
6
votes
2answers
244 views

Simple groups and irreducible characters of degree 3

I have posted this question on mathstack echange but did not get any answer. It mam trying my luck here. The only simple finite groups admitting an irreducible character of degree 3 are $\mathfrak{A}...
31
votes
3answers
2k views

Is there a nice explanation for this curious fact about cyclic subgroups?

Here's something that I noticed that quite surprised me. Let $G$ be a finite abelian group. Consider the following expression. $$ \nu(G) = \sum_{\substack{H \leq G \\ H \text{ is cyclic}}} |H| $$ It ...
5
votes
8answers
863 views

classification of $p$-groups

I have two questions regarding to $p$-groups. A $p$-group $G$ is said to be extraspecial of $G'=Z(G)$ has order $p$. Hence extraspecial groups are examples of $p$-groups with cyclic center. Of ...
2
votes
0answers
72 views

When is the restriction map $res:H^2(G,U(1))\to H^2(Z_p\times Z_p,U(1))$ not the zero map?

Consider $G$ to be a finite group with non-trivial Schur Multipler $H^2(G,U(1))$, where $G$ acts trivially on the circle group $U(1)$. By Example of a Schur-nontrivial group with no abelian subgroup ...
1
vote
1answer
160 views

Primes and the factorial

Conjectures: For all prime numbers $p>5$ there exist a prime number $q<p$ such that $q\equiv m!\!\pmod p$, for some $m$ with $2<m<p$. and For all primes $p$ there is a prime $...
3
votes
0answers
56 views

Group of Lie type as expanders: explicit estimates

In a paper Finite simple groups as expanders by M. Kassabov, A. Lubotzky and N. Nikolov there is a theorem, stating that there exists $\varepsilon>0$ and $k\in\mathbb{N}$, such that for every non-...
10
votes
5answers
710 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} (p^{n}-...
1
vote
2answers
213 views

Does the hyperoctahedral group have only 3 maximal normal subgroups?

An hyperoctahedral group $G$ is the wreath product of $S_2$ and $S_n$, where $S_{n}$ is the symmetric group on $n$ letters, or in other words the semi-direct product $G=S_2^n\rtimes S_n$, w.r.t. the ...
1
vote
3answers
153 views

p-groups and 2-generated abelian images

Let $p$ be a prime number. Is there a finite nonabelian $p$-group $G$ such that any finite epimorphic $2$-generated image of $G$ is abelian?
3
votes
1answer
169 views

representation of a group and its center

(I asked the following question at StackExchange but received no answer.) Let $G$ be a finite group and let $Z(G)$ be its center. Let $C=\mathrm{Rep}(G)$ be the category of finite dimensional ...
2
votes
0answers
109 views

Analogues of Goursat's Lemma for Infinite Products of Topological Groups

Recall that Goursat's Lemma has the following useful consequence. Let $G_1, G_2$ be finite groups with no common simple non-abelian quotients, and suppose $\gcd(|G_1^{\operatorname{ab}}|, |G_2^{\...
37
votes
5answers
2k views

How much of the ATLAS of finite groups is independently checked and/or computer verified?

In a recent talk Serre made some comments about proofs that rely on the classification of finite simple groups (CFSG) and on the ATLAS of Finite Groups. Namely, he said that a proof that relied on the ...
1
vote
0answers
154 views

The number of fixed points of an automorphism of $\mathbb{Z}_m\times\mathbb{Z}_n$

Let $m$ and $n$ be two positive integers such that the groups $\mathbb{Z}_m$ and $\mathbb{Z}_n$ have no common direct factor. Then an automorphism $f$ of $\mathbb{Z}_m\times\mathbb{Z}_n$ is of type $$\...
5
votes
1answer
103 views

Bounding the union of conjugates of a maximal subgroup of the Symplectic group over a finite field

Let $g \geq 1$ be a positive integer, and let $p$ be a prime. Consider the symplectic group $G := \operatorname{Sp}_{2g}(\mathbb{F}_p)$ of symplectic matrices with entries in $\mathbb{F}_p$. Let $M \...
8
votes
1answer
859 views

A dual version of a theorem of Øystein Ore in group theory

Let $(H \subset G)$ be an inclusion of finite groups. This post is a dual version for the Generalization of a theorem of Øystein Ore in which it's proved: Theorem: $\mathcal{L}(H\subset G)$ ...
4
votes
5answers
505 views

Intersection of all normalizers

This is probably standard for group-theorists: Let $G$ be a finite group. Is it true that the intersection of all normalizers of subgroups equals the center? If so, where do I find a proof? What about ...
11
votes
2answers
265 views

Finite groups with lots of conjugacy classes, but only small abelian normal subgroups?

Denote the commuting probability (the probability that two randomly chosen elements commute) of a finite group $G$ by $\operatorname{cp}(G)$. By a result of Gustafson [2], $\operatorname{cp}(G)=\...
4
votes
2answers
686 views

Cyclic subgroups of finite abelian groups

I learned from MO Subgroups of a finite abelian group that the problem of enumerating subgroups (not up to isomorphism) of finite abelian groups is a difficult one. Are there simple formulas if one ...
13
votes
1answer
338 views

Cohomology of central extensions of groups

Let G be a central extension of a finite group H by $Z/2$. I need an explicit description of the differentials $d_2$ and $d_3$ in the Lyndon-Hochschild-Serre spectral sequence which converges to the ...
3
votes
1answer
252 views

A question on $p$-central $p$-groups

Let $p$ be a fixed prime. A group $G$ is termed $p$-central if every element of order $p$ in $G$ lies in the center. Having a finite $p$-group $G$ of rank $k$ (the least integer, such that every ...
2
votes
1answer
91 views

Finite groups of planar homeomorphsims

Let $G$ be a finite subgroup of the group $H$ of orientation-preserving homeomorphisms of the plane that fix the origin. Is $G$ conjugate in $H$ to a group of rotations? I've been told this result ...
0
votes
2answers
723 views

Maximal subgroups of a finite p-group

I want to prove the following: Let $G$ be a finite abelian $p$-group that is not cyclic. Let $L \ne {1}$ be a subgroup of $G$ and $U$ be a maximal subgroup of L then there exists a maximal subgroup $...
4
votes
2answers
212 views

Exceptional isomorphisms between finite simple Chevalley groups

Steinberg's "Lectures on Chevalley Groups" https://math.depaul.edu/cdrupies/research/papers/chevalleygroups.pdf contain ``a complete list of isomorphisms" among the various finite simple Chevalley ...
0
votes
0answers
132 views

Ring structure on cohomology of groups

Assume that $G$ is a finite group and that $A$ is an arbitrary $G$-module. Then we know that can define the cohomology groups of $G$ with coefficients in $A$ in the usual way and we denote the latter ...
6
votes
3answers
482 views

Counting cyclic subgroups of order $p^{2}$: $p$ an odd prime vs. $p=2$

While playing with Frobenius' problem (about finite groups $G$ in which, for some positive integer $n \mid |G|$, there are exactly $n$ elements of order dividing $n$), I came up with the following ...
8
votes
1answer
755 views

Automorphism group of a finite group

I would like to ask if there exists an explicit description of $\mathrm{Aut}(G)$, the group of automorphisms of a finite group $G$, in particular, when $G$ is abelian. E.g., if $G = \mathbb{Z}/m\...
9
votes
1answer
620 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 ...
4
votes
3answers
737 views

solvable groups

Let $G$ be a finite group which has a cyclic maximal subgroup. Is $G$ solvable?
8
votes
2answers
267 views

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

Let $p$ be a prime number. For which finite $p$-groups $H$ is there a finite $p$-group $G$ such that $[G,G] \cong H$?
20
votes
1answer
933 views

Number of 2-dimensional irreducible representations of a finite group ?

Question: What is the number of two-dimensional irreducible representations of a finite group ? How it can be expressed in groups-theoretic terms ? (Number of 1-dimensional irreps is |G/[G,G]| ). ...
5
votes
2answers
513 views

Finite groups with a character having very few nonzero values?

A number theorist I know (who studies Galois representations) raised a question recently: Which finite groups can have an irreducible character of degree at least 2 having only $n=2, 3$, or 4 ...
6
votes
1answer
462 views

Non-vanishing of the Tate-Shafarevich kernel in group cohomology

Let $G$ be a finite group. Let $M$ be a finite $G$-module (a finite abelian group with an action of $G$). We consider a special kind of $G$-modules; in particular, our $M$ is a finite dimensional ...
4
votes
2answers
236 views

Is $(G\rtimes H,H)$ a Gelfand pair iff $G$ is abelian?

Let $G$ be a finite group and $K \subset G$ a subgroup. Then $(G,K)$ is a Gelfand pair if the double coset Hecke algebra $\mathbb{C}(K \backslash G / K)$ is commutative. Let $H$ be a subgroup of $...
4
votes
0answers
59 views

Is there an interval of finite groups, at index n, with strictly more elements than the subgroup lattice of any group, of order n?

Let $G$ be a finite group and $\mathcal{L}(G)$ its subgroup lattice. Let $s(n):= max\{|\mathcal{L}(G)| \text{ for } |G|=n \}$. There is an OEIS page for the sequence $s(n)$: A018216 1, 2, 2, 5, 2, ...
3
votes
2answers
143 views

Do the irreducible modules of this finite group preserve a tensor product structure?

I am interested in a particular group $G$, where $$ (A_4\times C_\ell) \lhd G \lhd S_4 \times D_\ell$$ Here, $C_\ell$ is cyclic, $D_\ell$ is dihedral of order $2\ell$, and the two inclusions both have ...
1
vote
0answers
110 views

On self-dual group-subgroup subfactors

Let $(N \subset M)$ be a finite index inclusion of ${\rm II}_1$ factors, and $N \subset M \subset M_1$ the basic construction. The subfactor $(N \subset M)$ is called self-dual if it is isomorphic to ...
5
votes
1answer
170 views

Subgroup ranks of the symmetric group

It's well known that every subgroup $G$ of $S_n$ has a generating set of size at most $n-1$ and that this generating set can be found algorithmically (by Jerrum's filter) I have heard many times a ...
1
vote
2answers
230 views

A good upper-bound for the cardinal of an interval of finite groups

This post is a relative version of General bound for the number of subgroups of a finite group Let $[H,G]$ be a interval of finite groups with $|G:H| = n$. Question: What is a good upper-bound of $|[...