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23
votes
1answer
898 views

Enumeration of a finite group

Let $G=\{g_1,g_2,...,g_n\}$ be a group with $e=g_1$ and $n$ is odd, Set $$a_1=g_1$$ $$a_2=g_1g_2$$ $$a_3=g_1g_2g_3$$ $$a_n=g_1g_2...g_n$$ I am looking for example that all $a_i$ are different from ...
11
votes
1answer
418 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 $p>g^{1\over 3}$ ...
2
votes
2answers
317 views

Finite Quotients and Resolutions of Singularities

So, I feel like I'm missing something obvious, but I have the following situation: Let $X\to Y$ be a finite group quotient of schemes (in fact, varieties) by the finite group $G$. Let $\tilde{Y}\to ...
1
vote
2answers
271 views

A semisimple group ring

Let $n \in \mathbb{N}$, $p$ a prime number, and $G$ a finite group of order coprime to $p$. Let $R = \mathbb{Z} /p^n \mathbb{Z}$ be the ring of integers mod $p^n$. Must $R[G]$ be semisimple? As noted ...
10
votes
2answers
270 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 ...
4
votes
2answers
350 views

Finite groups factorized into two simple alternating groups

My research is somehow related to the following question : Describe and classify all finite groups $G$ such that $G=HK$ with $H \cap K=1$, where $H \cong A_m$ and $K \cong A_n$ for some integers $m, ...
3
votes
1answer
155 views

Integral representations of groups of small order

I have a problem in which it would be helpful to know about the integral representations of some groups of small order (probably of fairly low degree). From what I've gathered so far, cyclic groups of ...
6
votes
1answer
276 views

What is the Grothendieck group of the category of $\mathbf{Z}_p[G]$-modules?

Let $G$ be a finite group. Let $\mathcal{O}$ be a suitably large finite extension of the $p$-adic integers, with residue field $\mathbf{F}_q$. The Grothendieck group of the category of ...
2
votes
2answers
168 views

Number of generators of the commutator

Can one find a function $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for every finite supersolvable group $G$ we have: $d(G') \leq f(d(G))$? Here $d(K)$ is the cardinality of a minimal set of ...
0
votes
0answers
74 views

Eigenvalues of the Cayley-like graph

Let $ F_q $ be a finite field of characteristic 2. Let $ x^2 + Sx +P \in F_q[x] $ be an irreducible polynomial over $ F_q $, and let $ g $ be one of its roots in $ F_{q^2} $. Define a map $ M: ...
3
votes
2answers
180 views

Elementary abelian $p$-subgroups of maximal rank in finite groups of Lie type

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $G$ be a reductive group defined over $\mathbb{F}_p$. For any $d\in\mathbb{Z}^+$, let $C_d(G)$ be the set of conjugacy ...
5
votes
2answers
222 views

Maximal order of finite subgroups of $GL(n,Z)$

I am interested in the finite subgroups of $GL(n,Z)$ of maximal order. Except for the dimensions $n = 2,4,6,7,8,9,10$ they are -- up to conjugacy in $GL(n,Q)$ -- in each dimension the group of signed ...
7
votes
0answers
187 views

A strong sum-product “for translates” in finite fields

In the course of some recent research, I've sketched out a proof of the following result. My basis question is: is the result interesting? Proposition There exists an absolute constant $c$ such ...
1
vote
0answers
174 views

A question on the poset of classes of isomorphic subgroups of finite groups

Given a finite group $G$, we consider the set $${\rm Iso}(G)=\{[H]\mid H\leq G\},$$where $[H]=\{K\leq G\mid K\cong H\}, \forall H\leq G$. Then ${\rm Iso}(G)$ can be partially ordered by defining ...
3
votes
0answers
187 views

Profinite groups, completions, and Schreier's formula

Let $G$ be a finitely generated profinite group, and $H \leq_o G$. We say that $H$ satisfies Schreier's formula in $G$ if $d(H) - 1 = (d(G)-1)[G:H]$. We say that $G$ satisfies Schreier's formula if ...
4
votes
2answers
306 views

Profinite completions

I call a profinite group $G$ Noetherian, if evrey ascending chain of closed subgroups is eventually stable. A standart argument shows that every closed subgroup of a Noetherian profinite group is ...
2
votes
1answer
124 views

a characterization for cyclic groups [duplicate]

Let $G$ be a group of order $n$ and its subgroup lattice be order-isomorphic to that of $\Bbb Z_n$. Is $G$ cyclic‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌?
0
votes
0answers
64 views

Fibres of square map on finite groups and the inverse set of a subset

Let $G$ be a finite group and $q:G\rightarrow G:g\mapsto g^2$ the square map. Now, if $A$ is a subset of one fibre of $q$, i. e. $$a^2=b^2$$ holds for all $a,b\in A$, is there always some $g\in G$ ...
27
votes
3answers
993 views

When is $S_n \times S_m$ a subgroup of $S_p$?

I asked the following question on math.stackexchange several months ago: Let $n,m,p>1$ be such that $S_n \times S_m \hookrightarrow S_p$. Does it imply that $p \geq n+m$? Derek Holt gave a ...
7
votes
1answer
357 views

Paths in groups

Given a finite group $G$, write $K(G)$ for the complete digraph on the elements of $G$. Label the edge from $g$ to $h$ by element $g^{-1}h$. Question: For what groups does there exist a Hamiltonian ...
1
vote
2answers
362 views

on the extensions of $ A_5$ by $A_5$ [closed]

Let $G$ be a finite group such that $G$ has a normal subgroup $H$ and $H$ is isomorphic to the alternating group $A_5$. Also we know that $G/H \cong A_5$. Can we say that $G \cong A_5\times A_5$? ...
6
votes
1answer
247 views

Fast construction of straight line programs?

Given a group $G$ and a set of generators $A$, we can ask ourselves (and do ask ourselves all the time) to bound the diameter of $G$ with respect to $A$. The diameter, let us recall, is defined to be ...
3
votes
1answer
142 views

Aschbacher's description of Sylow 2-subgroups

I'm looking for a few more examples of using Aschbacher (1980)'s “fundamental SL2 subgroups” description of Sylow 2-subgroups of finite groups of Lie type in odd characteristic. I do not yet ...
1
vote
1answer
65 views

A characterization of almost relatively free, finite $p$-groups

Let $G$ be a finite minimally $d$-generated $p$-group. If $G$ is relatively free, that is $G$ is a quotient of the free group $F$ on $d$ generators by a fully invariant subgroup of $F$, then the ...
2
votes
0answers
50 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 ...
3
votes
2answers
209 views

On the groups of order $p(p^2+1)$

Let $G$ be a group of order $p(p^2+1)$, where $p$ is an odd prime number and $p>3$. Easily we can see that $G$ is solvable and so $G$ has a Hall subgroup $L$ of order $p^2+1$. Also we know that ...
2
votes
0answers
71 views

“A locally dual polar space for the Monster”

I am currently looking at Ronan and Stroth's 1984 paper Minimal Parabolic Geometries for the Sporadic Groups. When considering the $3$-minimal parabolic system of $F_{1}$, they cite a preprint by ...
1
vote
0answers
123 views

Questions about some finite p-groups of coclass 2

I have two questions about a specific type of finite $p$-groups that i've seen in an interesting work on automorphisms of finite p-groups. Let $G$ a finite $p$-group of order $p^{n}$ such that: - ...
0
votes
1answer
190 views

Projective characters with corresponding factor set

The following is just a follow up to my previous question. I have a finite group $H$ with 14 ordinary characters. The Schur multiplier $M(H)\cong 2^2$. Hence the group $H$ will have 3 sets of ...
11
votes
0answers
269 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 ...
48
votes
3answers
2k views

How do I verify the Coq proof of Feit-Thompson?

I probably don't have the appropriate background to even ask this question. I know next to nothing about formal or computer-aided proof, and very little even about group theory. And this question is ...
6
votes
2answers
343 views

Sylow 3-subgroups of symmetric group

Is there any routine technique to find a set of permutations which generate a Sylow $3$-subgroup of the symmetric group $S_{n}$?
4
votes
2answers
550 views

Decomposing representations of finite groups

Let $G$ be a finite group, $p$ a prime number. We denote by $\mathbb{F}_p$ the field of cardinality $p$. Let $V$ be an infinite dimensional representation of $G$ over $\mathbb{F}_p$. Must there be ...
7
votes
1answer
397 views

Is there a precise notion of “almost all” such that almost all finite groups are Galois groups of extensions of the rationals?

I vainly tried to define a notion of "almost solvable group" such that every "almost solvable group" is the Galois group of a finite extension of the rationals, but I can't figure out the right way to ...
4
votes
1answer
166 views

Maximum length of chains of subgroup in GL(n,q)

Let G=GL(n,q) be a general linear group n-dimensional over a field with q element (q power of a prime). I am looking for an estimate of maximum length of chains of subgroup in G. Thanks.
2
votes
0answers
90 views

Coprime automorphisms of finitely generated pro-$p$ groups

Let $P$ be a finitely generated pro-$p$ group and let $G$ be a semidirect product $P \rtimes A$, where $A$ is a finite group of order coprime to $p$ that acts faithfully on $P$. Then one can show ...
2
votes
1answer
329 views

Groups like symmetric group

For sufficiently large n consider this question Let $G$ be a finite group with the following properties: $|G|=n!$ $H,K$ are subgroups of $G$ such that $H\cap K=1$ and $H\cong S_{3}$ and $K\cong ...
3
votes
0answers
102 views

Automorphisms of profinite groups

Let $d,n \in \mathbb{N}$, and $p$ a prime number. Let $F$ be a free pro-$p$ group on $d$ generators. Is there an automorphism of $F$ of order $n$?
2
votes
1answer
131 views

generality of the lattice of normal subgroups

Let $(X,\le)$ a (finite) modular lattice. Is there a (finite) group $G$ such that the lattice of all normal subgroups of $G$ is isomorphic to $(X,\le)$?
4
votes
1answer
284 views

Which finite groups can be characterized by their subgroup orders?

Given a finite group $G$, we denote by $\pi_s(G)$ the set of orders of its subgroups. Which finite groups $G$ can be characterized by the set $\pi_s(G)$, i.e. $\pi_s(H)=\pi_s(G)$ implies $H\cong G$? ...
6
votes
2answers
268 views

Subgroups from which all class functions extend to class functions on the ambient group

Until someone suggests better terminology, let me call a subgroup H of a finite group G segregated if every class function on H can be extended to a class function on G. Equivalently, H should have ...
23
votes
1answer
649 views

Properties to have matrices that commute in $\mathrm{GL}_n(\mathbb C)$

Let $G$ be a finite subgroup of $\mathrm{GL}_n(\mathbb C)$, $A,B \in G$ whose eigenvalues are thus in the unit circle. Assume that the eigenvalues ​​of $A$ are included in a circle arc of ...
0
votes
0answers
69 views

What algebras does the hidden subgroup problem for finite abelian groups apply to?

Shor's algorithm is said to solve the hidden subgroup problem for finite Abelian groups, of which factoring and the discrete logarithm problem for integers belong to. Apparently the HSP for finite ...
3
votes
2answers
246 views

Bound for the Frattini subgroup of a $p$-group

Assume that $G$ is a finite $p$-group, $p$ odd, with a non-trivial elementary abelian Frattini subgroup. Then both $\Phi(G)$ and $G/ \Phi(G)$ are vector spaces over $\mathbb{F}_p$. Is it possible to ...
4
votes
1answer
260 views

Normal Subgroup Growth

Let $F$ be a free group on $d$ generators. Denote by $F_{k}$ the $k$-th term in $F$'s derived series. Put $G = F/F_k$. What is the normal subgroup growth of $G$? Explicitly, for each natural number ...
4
votes
2answers
248 views

Simple groups and words

Let S be a finite simple nonabelian group, w a word in a finite number of variables which is not a power of another word. Must there be a substitution of elements of S in w such that the resulting ...
10
votes
0answers
299 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, ...
9
votes
0answers
241 views

Non split extension isomorphic (as a group) to a split extension

$\def\Z{\mathbb{Z}}$ Let $A$ be a finite abelian group and $G$ a finite group acting on $A$. Then the extension $0 \to A \to E \to G \to 1$ is splits if and only if the corresponding $2$-cocycle is ...
7
votes
0answers
254 views

Connection between two theorems on character values?

In a recent arXiv preprint here, Dipendra Prasad has revisited a 1976 theorem of Kostant (Theorem 2 in the paper On Macdonald's $\eta$-function formula, the Laplacian and generalized exponents, ...
2
votes
0answers
122 views

Two $p$-groups whose automorphism groups have isomorphic Sylow $p$-subgroups

Fix a prime $p$, and let $M$ be the unique nonabelian group of order $p^3$ and exponent $p$. Let us denote by $E_n$ the elementary abelian group of rank $n$. Is it true that $\operatorname{Aut}(M ...