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Questions on group theory which concern finite groups.

-1
votes
0answers
90 views

Finite groups with trivial center [on hold]

I know the Symmetric groups, Alternating groups and Frobenius groups of order $pq$, where $p$ and $q$ are distinct prime number, have trivial center. I want to know a classification of finite groups ...
1
vote
0answers
104 views

Representing curves using words

I am trying to understand how in this paper https://arxiv.org/abs/1412.0101 he represents curves with words. This is on page 10 of the paper. Assume that two piecewise smooth closed curves $\gamma_1$ ...
3
votes
1answer
76 views

Lower frattini subgroup

Let $G$ be a finite group. Define $\Phi_{-}(G)$ as the subgroup of $G$ generated by all the minimal subgroups of $G$ (a minimal subgroup of $G$ is a subgroup of $G$ of prime order). It is easy to ...
-1
votes
1answer
80 views

Is the numerable product of finite abelian groups a cantor set? [closed]

Today I have heard that statement but I can't find the reference. Can somebody know a reference and/or a proof?
7
votes
1answer
172 views

Finite group with a character having one nonzero absolute value

Let $G$ be a finite group. Assume that $\chi$ is a complex irreducible character of $G$ of degree $n\geq 2$, with the property that for each element $g\in G$ either $\chi(g)=0$ or $|\chi(g)|=n$. ...
5
votes
1answer
325 views

Question on Hall's theorem

Theorem 9.3.1 in Hall's group theory says: Let $G$ be a solvable group and $|G|=m\cdot n$, where $% m=p_{1}^{\alpha _{1}}\cdot \cdot \cdot p_{r}^{\alpha _{r}}$, $(m,n)=1$. Let $% \pi =\{p_{1},...,p_{r}...
2
votes
0answers
68 views

Width of symmetric groups

MSE crosspost For any (finite) group $G$ its length $l(G)$ is the length of maximal chain of proper subgroups (it's known and pretty widely used invariant). But we can also define width function $w_G(...
1
vote
0answers
110 views

about a strange property of p-groups of maximal class

I am trying to look for a finite $p$-group of maximal class of order at least $p^{2p+1}$ exponent at least $p^3$ which possibly has the following property : If s is an element in $G-G_1$ ($G_1$ is ...
1
vote
0answers
41 views

Weighted cancellation norm of a word computation

A symmetric set without identity $S$ is a set with a bijective function $inv : S \rightarrow S$ with no fixed points such that $inv(inv(x)) = x$ for any $x \in S$. We say that two disjoint pairs $\{...
1
vote
0answers
76 views

Certain $p$-group with cyclic center

Let $G$ be a finite $p$-group of derived length $d$, which is not a Dedekind group. (i.e., possesses at least one non-normal subgroup). Let $G^{(d-1)}$ be the unique normal subgroup of $G$ of order $...
4
votes
2answers
300 views

$G\cong C_4\times A_5$ or $C_2\times C_2\times A_5$?

Let $G$ be a finite group of order $240$. If $G\cong C_4\times A_5$ or $C_2\times C_2\times A_5$, then the all degrees of irreducible $\mathbb{C}$-characters of $G$ are $ [1,1,1,1,~3,3,3,3,3,3,3,3, ~...
7
votes
0answers
156 views

Integral representations of finite groups and lattice point geometry

This contains both a reference request, and a specific problem. Let $K$ be a finite group, and let $\theta: K \to {\rm GL}(d,{\bf Z})$ be a (faithful) group representation over the integers. Consider ...
7
votes
0answers
103 views

Small modules over finite group with large cohomology

Looking at this Example of group cohomology not annihilated by exponent of $G$? I stumbled upon one question I couldn't solve (probably because it's hard), so I post it here. Using Lyndon resolvent, ...
4
votes
0answers
171 views

Conjugacy class representatives for the automorphism group of a finite abelian group

Given a finite abelian group $A$, I'd like a list of conjugacy class representatives for its automorphism group ${\rm Aut}(A)$. In fact, it's not important that I have exactly one representative from ...
3
votes
2answers
343 views

Finite groups with small God's numbers

Let $G$ be a finite group and $S$ be generating set it. Now given all words with alphabet $S$, then there exists a minimum word length $N(S,G)$ such that all group elements are represented by a word ...
0
votes
1answer
156 views

Is $G$ non-solvable?

Let $G$ be a finite group of order $2^7\cdot3^3\cdot5^2\cdot7$. Let $\mathrm{Irr}(G)$ be the set of all the irreducible $\mathbb{C}$-characters. Suppose that (1) there is a character $\chi\in\mathrm{...
5
votes
0answers
60 views

A class function defined using Frobenius-Schur indicators

Let ${\rm Irr}(G)$ be the set of complex irreducible characters of a finite group $G$. The Frobenius-Schur indicator of $\chi\in{\rm Irr}(G)$ is defined to be $\epsilon(\chi):=\frac{1}{|G|}\sum_{g\in ...
1
vote
0answers
84 views

Character degrees of a finite group?

Let $G$ be a finite group, $R(G)$ be the solvable radical of $G$, and $G/R(G)$ be an almost simple group with socle $S\cong PSL_2(3^p)$, where $p$ is an odd prime. Also let $[G/R(G):S]=p$ and $\theta$ ...
11
votes
1answer
244 views

Genus of Cayley graph of $A_5$ with two generators of order 5

The Cayley graph of $A_5$ with two generators of order 5 seems rather complicated. I am wondering what is its graph genus (orientable or non-orientable). The best I could get by trial and error is an ...
4
votes
0answers
105 views

Does $G$ have a normal abelian Sylow $2$-subgroup?

Let $G$ be a finite group. Let $|G|=2^\alpha n$ where $(2,n)=1$ and $\alpha$ is a positive integer. Suppose that $\def\cd{\operatorname{cd}} n=\max \cd(G)$, and $n^2>\frac{1}{2}|G|$, where $\cd(G)$...
3
votes
0answers
62 views

Minimal dimension of faithful representation of $G(\Bbb F_q)$

Let $k=\Bbb F_q$ be a finite field, $\mathcal{G}$ be a reductive group over $k$, denote $m(G)$ by the minimal dimension of faithful representation of $G=\mathcal{G}(\Bbb F_q)$. Do we know the value $m(...
7
votes
1answer
492 views

On the structure of a finite group of order $144$

Let $G$ be a finite group of order $144$. Suppose there is an irreducible $\mathbb{C}$-character $\theta$ such that $\theta(1)=9$. QUESTION: Prove $G\cong A_4\times A_4$. By using Magma, we know ...
4
votes
1answer
142 views

Kantor's Singer cycle theorem

I'm trying to understand the proof of Kantor's Singer cycle theorem, which asserts that if $G$ is a subgroup of $\operatorname{GL}(n,q)$ containing a Singer cycle then $\operatorname{GL}(n/s,q^s) \leq ...
8
votes
1answer
263 views

Is a $G$-invariant character $\theta$ of $H$ extendible to $G$?

Let $G/H\cong PSL(2,11)$, and $\theta$ be an irreducible $\mathbb{C}$-character of $H$. Suppose $\theta$ is invariant in $G$ and $\theta(1)=9$. Question: Is $\theta$ extendible to $G$?
0
votes
0answers
113 views

Absolute center of a certain $p$-group $G$

Let $L(G)$ denote the absolute center of group $G$ that is the subgroup of $G$ consisting of those elements of $G$ which are kept fixed by every automorphism of $G$. We know that $L(G)$ is contained ...
6
votes
1answer
176 views

How much do we need to add to the generating set of the symplectic group to get $SL(2n,2)$?

Here $Sp(2n,\mathbb{F}_2)$ means the group of matrices preserving the form $\Omega = \left( \begin{array}{cc} 0&I \\ -I&0& \end{array} \right)$, i.e. the symplectic group over an even ...
1
vote
0answers
64 views

The degrees of ordinary characters of $PSp(2n,q)$ and $P\Omega O(2n+1,q)$

The finite simple groups $PSp(2n,q)$ and $P\Omega O(2n+1,q)$ have a same order, where $n\geqslant3$ and $q$ is odd. What are the degrees of the ordinary characters of these two groups? Thanks!!!
0
votes
0answers
103 views

Counting conjugacy classes with a subgroup of prime index

I am trying to understand the classical method of counting classes from Burnside's old book (Note E) (also clarified a bit by Vera-Lopez, Conjugacy classes in finite solvable groups, 1984) : $G$ is a ...
5
votes
1answer
158 views

Schur covers of affine 2-transitive groups

I am interested in Schur covers of minimal 2-transitive groups. A theorem of Burnside gives that every finite 2-transitive group is either almost simple or affine. In the time since, these groups have ...
1
vote
0answers
56 views

p-group of maximal class

I am trying to prove that if $G$ is a $p$-group of maximal class and order $p^4$ ($p$ odd), then its unique two-step centraliser $G_1=C_G([G,G])$ is of the form $C_{p^2}\times C_p$. It is clear from ...
1
vote
0answers
65 views

Example of Chevalley group

I'm trying to grasp my head around Chevalley groups. I could greatly benefit from an illustrating example here. Could somebody give an example of constructing a simple but nontrivial Chevalley group? ...
25
votes
2answers
795 views

Is the cohomology ring of a finite group computable?

Is there an algorithm which halts on all inputs that takes as input a finite group ($p$-group if you like) and outputs a finite presentation of the cohomology ring (with trivial coefficients $\mathbb{...
2
votes
1answer
117 views

Maximal subgroups of Alternating groups of degree $p$, for some prime $p$

Let $p\neq5$ be a prime number such that $q=(p-1)/2$ is prime. Does there exist an Alternating group of degree $p$ in which every minimal subgroup of order $p$ is contained properly in exactly one ...
3
votes
1answer
134 views

Computation of group homology $H_2 ((\mathbb{Z}/3\mathbb{Z}) \rtimes (\mathbb{Z}/4\mathbb{Z}),\mathbb{Z})$

In my research I need to compute the group homology of the dicyclic group Dic3, which is a semi-direct product of $\mathbb{Z}/3\mathbb{Z}$ and $\mathbb{Z}/4\mathbb{Z}$, and let's denote it by $G$, we ...
2
votes
0answers
55 views

A lattice ordered by inclusion and isomorphic to the lattice of quotient groups of a finite group

Let $G$ be a finite group. Consider the lattice $$L=\{ G/N:\text{$ N $ is a normal subgroup of $G $}\},$$ where $G/N \leq G/K$ if and only if $K\leq N$. The lattice operations ∧ and ∨ on quotient ...
5
votes
0answers
95 views

$m$-thick sets with small $n$-fold sumsets in finite cyclic groups

Problem. Is it true that for every positive integers $n,m$ there exists a subset $A_{n,m}$ of a finite cyclic group $G$ having the following two properties: $(\Sigma_n)$ the $n$-fold sum $A_{n,m}^{+...
4
votes
1answer
107 views

Central extensions of Suzuki 2-groups

Recall the definition of the finite Suzuki 2-groups: These are finite non-abelian 2-groups that contain more than one involution such that a solvable group of automorphisms permutes the involutions ...
5
votes
1answer
155 views

Large subgroups of $S_n$ without large symmetric or alternating subgroups

I'm interested in determining the existence of a permutation group $G\subseteq S_n$ of the following form. $G$ is large. Meaning that $G$ have at least $n!/2^{o(n)}$ elements. Equivalently, their ...
1
vote
0answers
46 views

Relation Among Conjugacy Classes

This is more a request to find out if there is any work in the literature discussing certain things. Is there a naturally defined partial ordering on the set of conjugacy classes of a finite group G? ...
4
votes
0answers
185 views

How many conjugacy classes of elementary abelian subgroups of rank $2$ does $GL_{n}(Z / pZ)$ have?

Let $p$ be a prime number and $G=GL_n ( \mathbb{Z} / p \mathbb{Z} )$. Consider the set $U$ of upper-triangular matrices of $G$ having entries of $1$ on the diagonal. The cardinality of $U$ is $p^{\...
6
votes
1answer
280 views

Group (Co)Homology of Symmetric Group

The question concerns the group homology or group cohomology of symmetric groups. The entries in groupprops.subwiki.org and in this MO post show the results for the symmetric group S$_4$. groupprops....
3
votes
0answers
80 views

does infinite leinster groups indicates infinite perfect numbers?

The simplest definition i managed to find is: Leinster group is a finite group which her order equal the sum of its normal subgroups order, and as a perfect number is a positive integer that is equal ...
1
vote
1answer
166 views

A question on the number of involutions in a 2-group

Is there a classification of groups of order $2^n$ and exponent $4$ containing exactly $2^{n-1}-1$ involutions?
3
votes
1answer
83 views

cohomology of finite groups of lie type with coefficients in the adjoint module

Let $\mathbb G$ be a connected, semisimple, split group over a finite field $\mathbb F_q$ and let $G = \mathbb G(\mathbb F_q)$. Let $\mathfrak g$ be its Lie algebra, an $\mathbb F_q$-vector space with ...
1
vote
0answers
113 views

Conjugacy classes of non-normal subgroups of a finite $p$-group

Let $G$ be a finite $p$-group of derived length $d$ and nilpotency class $c$. Suppose that $G$ is not a Dedekind group (i.e., possesses at least one non-normal subgroup). Suppose that $G^{(d-1)}$ has ...
16
votes
1answer
698 views

Why do these two Monster-related calculations yield $163$?

Fact 1: (1979, Conway and Norton)$^{1}$ "There are $194-22-9=\color{blue}{163\,}$ $\mathbb{Z}$-independent McKay-Thompson series for the Monster." Note: There are 194 (linear) irreducible ...
1
vote
0answers
79 views

$G$-invariant bilinear maps

Let $G$ be a finite group and $M$ a finitely generated $\mathbb{Z}G$-module. Then $M \otimes k$ is a representation of $G$ for any field $k$. I am interested in the number of ways we can turn $M \...
0
votes
0answers
29 views

What is an example of a constructive encoding of binary strings modulo an arbitrary permutation group $G$?

Given a group $G \leq S_n$ we can construct by the axiom of choice a function $f : \{0, 1\}^n \rightarrow \{0, 1\}^{ceil(log_2 |\{0, 1\}^n/G|)}$ such that for any orbit $O$ of binary strings under $G$,...
3
votes
1answer
170 views

Looking for example of quotient of group algebra by ideal of group ring which fails to be injective

I am looking for an example of a group ring $\mathbb{Z}[G]$ of a finite group $G$ along with a lattice $I$ (in the case at hand the word 'lattice' means: a $\mathbb{Z}[G]$-submodule which is ...
3
votes
2answers
177 views

Length of composition series in a primitive group

Let $G$ be a primitive group acting on a set $\Omega$ with $n$ elements. By Cameron/Liebeck (essentially a consequence of the Classification + O'Nan-Scott), there are two possibilities: (a) $G$ has a ...