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Finite group and cyclic cover

Suppose the finite group $N$ surjects to finite group $F$. It is true that for any $G = N ⋊_α \mathbb{Z}$ there are infinitely many covers of $G$ that are cyclic and surject to $F$. But is this ...
Ma Joad's user avatar
  • 1,755
3 votes
0 answers
170 views

What is $G_2(2^m)$, and how is it embedded in $\Gamma L_6(2^m)$?

I am trying to understand the classification of doubly transitive groups, specifically the nonsolvable affine case. Dixon and Mortimer (p.244) says there are three infinite families, one of which is $\...
Dustin G. Mixon's user avatar
3 votes
0 answers
62 views

Torus in the small Ree group ${}^2G_2$ over an infinite field

In “Simple group of Lie type” by R. W. Carter there is a remark (after Theorem 13.7.4): It is not known whether $H^1$ coincides with the set of $\sigma$-invariant elements of $H$ if $\mathfrak{L}$ ...
Andrei Smolensky's user avatar
29 votes
2 answers
1k views

Quillen + construction for finite groups

Is there an example of two non isomorphic finite groups $G$ and $H$ such that $BG^{+}$ is homotopy equivalent to $BH^{+}$ ?
mathphys's user avatar
  • 1,629
7 votes
0 answers
229 views

Computation of $H^2(S_n,\mathbb{Z}/2\mathbb{Z})$

Do you have a nice modern reference where I could find the computation of $H^2(S_n,\mathbb{Z}/2\mathbb{Z})$, where the action is trivial ? I have looked at the very few books on cohomology of groups ...
GreginGre's user avatar
  • 1,766
0 votes
1 answer
227 views

Finite group cohomology with roots of unity as coefficients

Let $G$ be a finite group of order $n$, and let $L := (\mathbb{Q}/\mathbb{Z})^d$ be a (not necessarily trivial) $G$-module (we assume that $d$ is finite). By a direct limit argument, there must be a ...
Pablo's user avatar
  • 11.3k
10 votes
1 answer
1k views

Maximal order of elements in SL(n,q)

The maximal order of an element of $\mathrm{GL}(n,\mathbb{F}_q)$ is $q^n-1$, where the characteristic of $\mathbb{F}_q$ is odd $p$. See here for a nice proof that uses the Cayley-Hamilton Theorem. ...
Sean Lawton's user avatar
  • 8,529
6 votes
1 answer
366 views

Group of order $5p^aq^b$

In Lectures by Dan Bump on Modular representation theory, Theorem 13.14 states that whenever $G$ is a non-abelian simple group of order $|G|=p^aq^br$ for distinct primes $p$,$q$, and $r$, every $r$-...
FunctionOfX's user avatar
5 votes
2 answers
530 views

Wielandt automorphism tower theorem

I wanted to know if anyone can point me to an (ideally freely available) english translation of the proof of Wielandt's Automorphism Tower Theorem (1939). The theorem states the following: Given a ...
Justin Benfield's user avatar
9 votes
2 answers
762 views

Solutions of $x^d=1$ in the symmetric group

L Moser and M Wyman, On solutions of $x^d = 1$ in symmetric groups, Canad. J. Math., 7 (1955), pages 159-168, explored asymptotic behavior of the cardinality of such permutations: $$f_d(n):=\#\{\pi\in\...
T. Amdeberhan's user avatar
6 votes
1 answer
205 views

Understanding (statement of) a theorem of Jack McLaughlin

In the book Twelve sporadic groups, Griess states If $A$ is an abelian group, $G$ acts on $A$, $z\in Z(G)$ satisfies $z-1\in$ Aut$(G)$, then $H^n(G,A)=0$ for $n\geq 0$. This is an observation of ...
Soluble's user avatar
  • 1,169
1 vote
0 answers
150 views

Reference request for the list of nilpotent subgroups of SU(2)?

It's not hard to show that all non-abelian nilpotent subgroups of $SU(2)$ are actually finite and in fact are conjugate to one of the generalized quaternion groups of order a power of two, $$Q_{2^n} =...
Omar Antolín-Camarena's user avatar
3 votes
0 answers
269 views

Reference for the rank of the BN-pair of the finite simple groups of Lie type and not Chevalley

The rank of the BN-pair of a Chevalley group is the number of simple roots of its Lie algebra, which is the index of the name of its Dynkin diagram ($A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4,G_2 $). ...
Sebastien Palcoux's user avatar
9 votes
1 answer
2k views

Finite groups in which all proper subgroups are cyclic

Is there any classification of finite group in which all proper subgroups are cyclic? Would you please tell me a reference?
benyamin's user avatar
2 votes
1 answer
197 views

Hall $\pi$ subgroups that controls its own fusion

Theorem:Let $P\in Syl_p(G)$ for a finite group $G$. Then $G$ has a normal $p-$ complement if and only if $P$ controls its own fusion. I wonder if similar argument is true for Hall subgroups (in ...
mesel's user avatar
  • 1,169
2 votes
1 answer
242 views

Products of groups with three generators

In this question it mentions how Jesse Douglas used a Zappa-Szep product to classify some finite groups with two generators, and others did the same thing with infinite cyclic groups. I've been ...
Alex Meiburg's user avatar
  • 1,203
7 votes
1 answer
313 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 ...
Joe Bebel's user avatar
  • 539
3 votes
2 answers
315 views

Character kernels in the lattice of subgroups of a finite abelian group

I am looking for any efforts that have been made to characterize the character kernels (equivalently, the subgroups yielding cyclic quotients) inside the lattice of subgroups of a finite abelian group....
benblumsmith's user avatar
  • 2,851
3 votes
2 answers
337 views

Frobenius Groups of Automorphisms

Recently, I am looking different papers on the topic $$\mbox{Frobenius groups of automorphisms of a group.}$$ But I am familiar with Frobenius groups only; not with their (faithful) actions on groups. ...
p Groups's user avatar
  • 261
3 votes
2 answers
469 views

Maximal size of minimal generating set

Let $G$ be a finite group. Denote by $D(G)$ the maximal size of a minimal generating set, (minimal in the sense of inclusion). I vaguely remember seeing recently something on $D(G)$. Can anyone refer ...
Yiftach Barnea's user avatar
2 votes
0 answers
190 views

Expository papers for Feit–Thompson Theorem [duplicate]

Feit–Thompson theorem states that every finite group of odd order is solvable. Its proof is near 300 pages so it is definitely not an easy paper to read without a prior knowledge about the general ...
user avatar
3 votes
1 answer
309 views

Intersection of maximal subgroups of PSL(2,q)

Let $G := PSL(2,2^r)$, and let $M$ be a maximal subgroup of $G$ isomorphic to $PSL(2,2^s)$. I need to compute $H := M \cap M^g$ for $g \in G-M$. It seems to me that $|H|$ must be $2^r, 2^r\pm 1$ or $...
Amin's user avatar
  • 307
3 votes
0 answers
186 views

Which Dihedral Groups are $\text{CI}$-Groups?

Let $D_{n}$ denotes the dihedral group of order $2n$. Firstly, for self-referencing of the question, I give some definitions which are standard. Let $G$ be a finite group. A subset $S$ of group $G$ ...
Shahrooz's user avatar
  • 4,784
7 votes
1 answer
169 views

What is the maximal possible rank of a subgroup of a special linear group mod a prime?

Let $p$ be a prime number, and let $\mathbb{F}_p$ be the unique field of cardinality $p$. What is $\max \{d(H) : H \leq \mathrm{SL}_3(\mathbb{F}_p)\}$? Here we denote by $d(G)$ the smallest ...
Pablo's user avatar
  • 11.3k
2 votes
0 answers
187 views

Classification of Automorphism set of a Regular graph

Let $A$ be the adjacency matrix of an $r$-regular graph $G$ with $n$ vertices (Not complete or cycle graph) . Also, let $Aut(G)$ be the set of all its automorphisms (i.e. set of permutation matrices)....
Michael's user avatar
  • 267
6 votes
4 answers
561 views

SO$(4)$ (& SO$(n)$) characterization?

I believe it is the case that any finite subgroup of SO$(3)$ (the $3 \times 3$ orthogonal matrices of determinant $1$) is either a cyclic group $C_n$, or a dihedral group $D_n$, or one of the groups ...
Joseph O'Rourke's user avatar
2 votes
1 answer
159 views

Counting elements with certain word length in abelian groups

Given a (finite) abelian group $G = \langle S \mid R \rangle$, has the problem of counting the number of elements which can be expressed as a word (in $S$) of length $\leq k$ been studied? If so, ...
Calle's user avatar
  • 121
16 votes
4 answers
1k 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 ...
Matt Samuel's user avatar
  • 2,168
3 votes
0 answers
102 views

Localized at $p$ integral representations of finite elementary $p$-groups

Let $C_p$ be a cyclic group of prime order $p$. Let $F=C_p^n=C_p\times\dots\times C_p$ ($n$ times). I would like to to classify finite dimensional representations of $F$ over ${\mathbb{Z}}$. However, ...
Mikhail Borovoi's user avatar
30 votes
1 answer
592 views

Guess that group via product queries

Suppose someone (person B) knows a finite group $G$ of order $n$. You (person A) know only the order $n$, and that $1$ is the name of the identity element. The group elements are named $1,2,\ldots,n$ ...
Joseph O'Rourke's user avatar
3 votes
0 answers
282 views

Galois correspondence subgroups/subsystems

In this paper (1998) by M. Izumi, R. Longo, S. Popa, there is the following result (page 49) on compact groups: Lemma 3.16. Let $G$ be a compact group and $Rep(G)$ the category of finite ...
Sebastien Palcoux's user avatar
4 votes
0 answers
135 views

Improvements of the Reidemeister-Schreier index formula for particular classes of groups

I have a couple of questions regarding possible improvements of the Reidemeister-Schreier index formula: let $G$ be a $d$-generated group and let $H$ be a subgroup of $G$, then $$d(H) \le (d-1) \...
Matteo Vannacci's user avatar
6 votes
1 answer
1k views

Structure of symplectic group over finite fields

We are working over the finite field $\mathbb{F}_{q}$ of odd prime characteristic $p$ and of cardinality $q$ some power of $p$. We recall the symplectic group $Sp(4,\mathbb{F}_{q})$ as the group of ...
Silam's user avatar
  • 85
10 votes
2 answers
815 views

Paper by I. N. Sanov, Solution of the Burnside problem for exponent 4

I have searched extensively online and for copies of printed journals containing the paper which details Sanov's solution to the Burnside Problem for exponent 4, which is widely cited in many papers ...
user50229's user avatar
  • 201
3 votes
4 answers
756 views

Lucido's three prime lemma

Let G be a finite solvable group. If p,q,r are distinct primes dividing |G|, then G contains an element of order the product of two of these three primes. This is lucido's three prime lemma. I ...
Bhaskar Vashishth's user avatar
8 votes
4 answers
658 views

Normal Covering of a Finite Group

Suppose $G$ is a finite group and $N_1, N_2, \cdots, N_k$ are proper normal subgroups of $G$. The set $\{ N_1, \cdots, N_k\}$ is called a normal cover for $G$, if $G = \cup_{i=1}^kN_i$. I need to the ...
Fatemeh Moftakhar's user avatar
0 votes
0 answers
192 views

Thin profinite groups - nonabelian analogues of p-adic integers

Let $p$ be a prime number, $S = C_p$ a cyclic group of order $p$, $G = \mathbb{Z}_p$ the profinite additive group of $p$-adic integers. It is well known that all the closed nontrivial subgroups of $G$ ...
Pablo's user avatar
  • 11.3k
0 votes
1 answer
215 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?
user48652's user avatar
6 votes
3 answers
964 views

Union of conjugates of a subgroup

Let $G$ be a finite group, $H \leq G$ a proper subgroup. It is well known that the union of the conjugates of $H$ does not cover $G$. I would like to know of more precise results (even in special ...
Pablo's user avatar
  • 11.3k
5 votes
2 answers
564 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, ...
Farrokh Shirjian's user avatar
3 votes
0 answers
127 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 ...
dward1996's user avatar
  • 295
27 votes
1 answer
1k 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 length $<\...
user avatar
6 votes
1 answer
435 views

Doubly primitive groups with simple socle

The classification of doubly transitive groups with simple socle is known. A good account of such classification can be found for example in this paper: Cameron, Peter J. Finite permutation groups ...
Leandro Vendramin's user avatar
19 votes
2 answers
943 views

Reference for the triple covering of A_6

I would like to ask for a reference (book, paper ...) for the following nice construction, which I have found as an exercise in some notes of a course by R. Borcherds. For $n=6$ or $7$ (and only in ...
abx's user avatar
  • 38k
17 votes
1 answer
575 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, ...
Kevin Walker's user avatar
  • 12.8k
9 votes
1 answer
3k 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\...
Hair80's user avatar
  • 675
9 votes
2 answers
701 views

Reference for embeddings of reflection groups (related to folding ADE Coxeter graphs)?

There are a couple of indirect methods, using Lie theory or Springer's general theory of regular elements in (real, complex) reflection groups, to construct natural embeddings among certain Weyl ...
Jim Humphreys's user avatar
2 votes
1 answer
241 views

Subgroup structure of orthogonal groups of small dimension over finite fields

How much is known about the subgroup structure of the orthogonal groups (of dimension n<=7, say) over finite fields? Can anyone point me in the direction of a good reference? I'm aware of a book by ...
MRD1729's user avatar
  • 393
3 votes
1 answer
118 views

On Groups of Maximal Class: Reference

I will be happy if one gives references (oncluding current research) for `classification' (structure) of $p$-groups of maximal class which contain abelian maximal subgroup (i.e. abelian subgroup of ...
Soluble's user avatar
  • 1,169
3 votes
1 answer
213 views

Groups with special automorphism group

I am looking for all finite groups $G$ such that for each subgroup $H$ of $G$ and each automorphism $\sigma$ of $H$ there exists an automorphism $\psi$ of $G$ whose restriction to $H$ is $\sigma$. Is ...
majid arezoomand's user avatar