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3
votes
1answer
43 views

Equivalence classes of (2,3)-pairs in PSL(2,q)

This may not be a research-level question, which is why I submitted it to math.stackexchange first, but so far the question there has barely been viewed, let alone answered. Apologies if submitting it ...
3
votes
0answers
99 views

Restricting the Steinberg representation of $SL_{2n}$ over a finite field to the symplectic group

Let $\text{St}_n(\mathbb{F}_q)$ be the Steinberg module (over $\mathbb{C}$) for $\text{SL}_n(\mathbb{F}_q)$. What is the irreducible decomposition of the restriction of $\text{St}_{2n}(\mathbb{F}_q)$ ...
2
votes
1answer
129 views

Is inner product preserved only by the stabiliser in a finite reflection group?

Is the following statement true for finite reflection groups? Let $G$ be a finite reflection group acting on $\mathbb{R}^n$, let $x, y\in \mathbb{R}^n$ and let $z$ be in the orbit of $y$. If ...
1
vote
0answers
100 views

An epimorphism into a profinite group

Let $p$ be an odd prime number, $G$ a finitely generated nonabelian profinite group, $L \lhd_o G$ a pro-$p$ group with $[G : L] = 2$. Suppose that there is a continuous surjection from $G$ onto a free ...
3
votes
1answer
85 views

A question about minimal nonnilpotent groups

Let $G$ be a minimal nonnilpotent group with cyclic Sylow $p$-subgroup $P$ and normal Sylow $q$-subgroup $Q$ [see Huppert, Endlich Gruppen I]. If $Q$ is abelian and $q > 2$, then can we get that ...
8
votes
1answer
163 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 $\mathcal{L}$ ...
0
votes
0answers
69 views

Dense free subgroups

Let $F$ be a free pro-$p$ group (for a prime number $p$) on a finite set $X$, $\Phi$ the abstract subgroup generated by $X$, $\{1\} \neq N \lhd_c F$. Is it possible that $\Phi \cap N = \{1\}$?
1
vote
0answers
44 views

Lower central series in a free pro-p group

Let $F$ be a nonabelian finitely generated free pro-$p$ group, $H \leq_c F$ of infinite index. Denote by $\{F_n\}_{n \in \mathbb{N}}$ the lower central series of $F$, and set $r_n = [F : F_nH]$. Is ...
0
votes
0answers
65 views

A bound on the size of the center

Let $p$ be a prime number, $F$ a free pro-$p$ group, $H \leq_c F$ of infinite index. Can it be that $$\sup_{N \lhd_o F} |Z((F/N)/C_{F/N}(HN/N))| < \infty ?$$
0
votes
0answers
92 views

Golod Shafarevich Inequality and Inequalities among higher Cohomology groups

As a consequence of Golod- Shafarevich, we get an inequality between second cohomology group of a $p$-group with coefficients in $F_p$ and the first cohomology group of a $p$-group with coefficients ...
0
votes
0answers
62 views

A normal form theorem for presentations of finite $p$-groups of nilpotency class $2$?

When constructing examples of nonabelian finite $p$-groups with abelian automorphism group (and certain other desired properties), the authors of papers like http://arxiv.org/pdf/1304.1974v1.pdf leave ...
-1
votes
0answers
68 views

Confusion about the projected component in an irreducible space in the tensor product decomposition using Littlewood-Richardson?

The regular representation of the symmetric group can be formulated in terms of an abstract tensor, where the action of the symmetric group elements is by means of permuting the indices. Given an ...
10
votes
2answers
490 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 ...
11
votes
1answer
529 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
274 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, ...
10
votes
2answers
262 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 ...
0
votes
0answers
98 views

Soluble group algebras and centralizers

Let $K$ be field with $\mathrm{char}(K) = p > 0$ and $G$ a finite group such that $KG$ is soluble. Then the $p$-Sylow-subgroup of $G$ is normal and contains the derived group of $G$ and every ...
8
votes
2answers
274 views

Sums of degrees of irreducible complex characters

The sum of the degrees of the irreducible complex characters (not the square sum which is the group order) is relevant to determine the dimension of a maximal torus of the Lie algebra associated to ...
1
vote
0answers
51 views

breadth of a finite p-group

The breadth of an element $x$ in a finite $p$-group G is defined to be that integer $b = br(x)$ such that $p ^b = |G : C_ G (x)|$, while the breadth $br(G)$ of $G$ is the supremum of $\{br_ G (x) | x ...
2
votes
1answer
146 views

Classification of indecomposable inclusions $(H \subset G)$ with $G$ decomposable

Definition: A group $G$ is indecomposable if: $G = G_1 \times G_2 \Rightarrow \exists i \ G_i = 1$. We can generalize the notion of indecomposable from groups to inclusion of groups as ...
7
votes
1answer
430 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 ...
2
votes
0answers
235 views

Prime order elements in $GL(n,\mathbb{Z})$

What is known about the elements of prime order $p$ in $GL(n,\mathbb{Z})$? All I know at this point is that $GL(n,\mathbb{Z})$ has an element of prime order $p$ iff $p-1\leq n$ and that there are only ...
2
votes
1answer
189 views

Irreducible representation of Heisenberg group with characteristic 2?

As we all know that the irreducible representation for Heisenberg group can be classified easily when the group is over a finite field $\mathbb{F}_q$, where $q=p^n$ and $p$ is a prime greater than ...
5
votes
0answers
64 views

Uniqueness of the direct product decomposition of inclusions of finite groups

This post is a generalization of Uniqueness of the direct product decomposition of finite groups. Here we look inclusions of finite groups $(H \subset G)$ instead of just finite groups. Definition: ...
0
votes
1answer
81 views

Absolute irreducibility of a symmetric square?

This is a question I received today by email, which somebody more experienced with finite group representations can probably answer directly. Take $F:=\mathbb{F}_q$ for some prime power $q$, so ...
3
votes
1answer
415 views

Existence of finite nonabelian groups satisfying certain identities

Is there a finite nonabelian group satisfying all of the following identities? $$ (x^py^p)^2 = (y^px^p)^2, \quad p = 2,3,5,7,11,\ldots (\text{primes}) $$ I thank you all in advance.
1
vote
1answer
105 views

The coproduct on the 2-boxes space of the goup-subgroup subfactor planar algebras

Let $(H \subset G)$ be an inclusion of finite groups. Let the subfactor $(\mathcal{R} \rtimes H \subset \mathcal{R} \rtimes G)$ with $\mathcal{R}$ the hyperfinite ${\rm II}_1$ factor, and its planar ...
0
votes
1answer
114 views

Questions on invariant operators of finite group representations

1) Is there an equivalent of the Casimir operator for an irreducible representation of a finite group? 2) Given an invariant operator of a certain group, can I check if it is invariant under only ...
6
votes
2answers
178 views

Asymptotic density of finite abelian and solvable groups

For every natural number n, let: Gn be the number of distinct group structures with at most n elements; An be the number of distinct abelian group structures wit at most n elements; Sn be the number ...
3
votes
0answers
242 views

What's the ratio of inclusions of finite groups with a distributive lattice?

Definition: Two inclusions of finite groups are equivalent, $(A \subset B) \sim (C \subset D)$, if: $(A/A_B \subset B/A_B) \simeq (C/C_D \subset D/C_D)$ with $A_B$ the normal core of $A$ in $B$. ...
5
votes
1answer
168 views

Special linear groups contained in symplectic groups

Let $q$ be a power of prime $p$, and $n, m, k$ positive integers such that $mk=2n$ and $2\leq m<2n$. Let $\mathrm{Sp}(2n,q)$ be the symplectic group of dimension $2n$ over $\mathrm{GF}(q)$ and ...
4
votes
1answer
168 views

orthogonal group in characteristic 2

Let $O(2,\mathbb{Z}_2)$ be the orthogonal group of order two matrices. On $\mathbb{Z}_2$ there should exist just one odd quadratic form, hence the stabilizer subgroup $O^-$ of an odd quadratic for ...
1
vote
1answer
117 views

Successive Schur covers

Let $G_0$ be a finite group and $G_j$ a Schur cover of $G_{j-1}$ for $j=1,2,3\ldots$. Is $G_2$ equal to $G_1$? If not, will the sequence stop after finite steps in general?
1
vote
0answers
101 views

Combination of two recent problems about finite groups of square orders

Is there any finite group $G$ of order $n=m^2$ satisfying the following conditions? (a) There is no subgroup of order $m$; (b) There exist subsets $A$, $B$ such that $|A|=|B|=m$ and $G=AB$. ...
0
votes
3answers
138 views

Possible degrees of faithful projective representations of $\mathrm{PSL}(k,q)$ and $\mathrm{Sp}(2k,q)$ over complex numbers

Let $q$ be a prime power and $k$ a positive integer. What are the possible degrees of faithful projective representations of the projective special linear group $\mathrm{PSL}(k,q)$ (over the Galois ...
10
votes
4answers
311 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 ...
3
votes
1answer
149 views

A question on conjugacy classes of central involutions in a finite group

An involution $a$ of a group $G$ is called central if there exists a sylow $2$-subgroup $H$ of $G$ such that $a \in C_G(H)$, or equivalently if the centralizer of $a$ contains a sylow $2$-subgroup. ...
6
votes
1answer
221 views

Generalization of Frobenius groups

Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some non-trivial element fixes a point. In other words, if in a ...
5
votes
0answers
108 views

Self-duality of the subgroup lattice of $G\times H$

Let $G$ and $H$ be finite groups and $\gcd(|G|,|H|)=1$. Suppose the lattice of all subgroups of $G\times H$ is self-dual. Is the lattice of all subgroups of $G$ self-dual?
2
votes
1answer
129 views

Classification of 2-groups with center of index 4

Can one obtain a classification of 2-groups with center of index 4, analogous to the classification of 2-groups with derived subgroup of index 4?
2
votes
0answers
122 views

Are there workable numerical approaches for the pentagon equation?

Warning: this post is the "numerical" analog of Are there workable algebraic geometry approaches for the pentagon equation? I've replaced "algebraic geometry" by "numerical" in its content, ...
0
votes
3answers
169 views

quasiprimitive non-solvable groups

I'm looking for the reference about quasiprimitive unsoluble groups. Actually we can find a lot of useful things about quasiprimitive solvable groups in "Representations of solvable groups by Manz and ...
9
votes
1answer
330 views

Factorization of a finite group by two subsets

I want to write a GAP program for checking the following question. Let $G$ be a given finite group with order $n$. Is it true that for every factorization $n=ab$ there exist subsets $A$ and $B$ ...
3
votes
1answer
242 views

Generating finite groups

Let $G$ be a finite group possessing a generating set of order $n \in \mathbb{N}$. Let $H \leq G$ and $x_1, \dots, x_n \in G$ for which $\langle H, x_1, \dots, x_n \rangle = G$. Must there be $h_1, ...
6
votes
1answer
552 views

Are there workable algebraic geometry approaches for the pentagon equation?

A pentagon equation is a system of polynomial equations of degree $3$ with several variables and integer coefficients, given by a fusion ring. A fusion ring is given by a finite set of integer ...
13
votes
0answers
474 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, ...
5
votes
0answers
80 views

representable functions on FinGrp

We look at functions $f: \text{FinGrp} \rightarrow \mathbb{N}$ such that $f$ is constant on isomorphism classes. Let's say that $f$ is representable if there is a (possibly infinite) group $K$ such ...
0
votes
1answer
190 views

Where can I find the classification of groups of order 16p? [closed]

I need to classify the groups of order $16p$ by their generators and relations between the generators. Can I find this classification anywhere?
1
vote
1answer
128 views

Products of subgroups that generate a finite group

Consider the following general problem. There is a finite group $G$ and $H_1,H_2 < G$. Suppose we know that $\langle H_1, H_2 \rangle = G$, i.e. $G$ is generated by $H_1$ and $H_2$. Denote by $n_0$ ...
5
votes
0answers
212 views

A particular example of solvable group

Let be $m$ an integer, $m>0$, $n=p_1^{a_1}...p_t^{a_t}$. Define $f(n)=a_1+...+a_t$. Let $G$ be a finite group and define $f(G)=\max\left\{\,f(\lvert g\rvert):g\in G\,\right\}$. Is there a finite ...