Questions on group theory which concern finite groups.

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77 views

### Finding Bijection between Permutation Set [on hold]

$\beta$ is a subset of symmetric group $S_n$ which acts on $n$ elements of set $X$. Permutations of
$\beta$ acts on $k $ elements of $X$ only.
Set $L$ is a set of $n$ labels which labels ...

**2**

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**0**answers

77 views

### Nonvanishing of the dual Euler totient on boolean intervals of finite groups

The rank $n$ boolean lattice $B_n$, is the subset lattice of $\{1,2, \dotsm n \}$.
Let $[H,G]$ be a boolean interval of finite groups. Its Euler totient is defined by $$\varphi(H,G):=\sum_{K \in ...

**4**

votes

**1**answer

85 views

### Primary invariants

This question is related to the earlier question which is in the given link:
Primary invariants of a finite group
Let $G$ be a finite group and $V$ a complex representation of degree $n$, and let ...

**2**

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**0**answers

95 views

### Chief factors and local formation

Every thing below is concerned with finite groups.
My question is about this paper
A class of groups is a collection $\mathcal{X}$ of groups with the property that
if $G \in \mathcal{X}$ and if $H ...

**9**

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**0**answers

272 views

### Finite groups inside an infinite group with the same homology

Suppose we have a triple of groups $G,H,K$ verifying the follwing conditions
$G$ and $H$ are finite groups and $K$ an infinite group.
there exists two monomorphisms $G\rightarrow K\leftarrow H$ ...

**2**

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**0**answers

79 views

### Computing characters of $\alpha$-projective representations

Given a finite group $G$, a finite cyclic group $A$ (viewed as a subgroup of $\mathbb{C}^{\times}$, i.e. generated by a $|A|$-th root of unity), and a 2-cocycle $\alpha\in Z^{2}(G,A)$. Recall that an ...

**1**

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**1**answer

81 views

### Primary invariants of a finite group

For a finite group $G$ and complex representation V of degree $n$, I would like to know the precise definition of Primary invariants. Does any set of n algebraically independent homogeneous invariants ...

**5**

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**1**answer

106 views

### Quotient of Coxeter complex in terms of double cosets?

In Victor Reiner's Quotients of Coxeter Complexes and $P$-Partitions, we have the below definition for the quotient complex of a Coxeter complex by a finite subgroup of the Coxeter group. I think ...

**10**

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**1**answer

159 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], ...

**5**

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**1**answer

296 views

### Known results in the Cohomology of finite groups

I am learning to compute cohomology of finite groups and came across this survey article http://www.ams.org/notices/199707/adem.pdf "Recent Developments in the cohomology of finite groups" by ...

**3**

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**0**answers

145 views

### For $P$ $\mathbb{Z}G$-projective, $\mathbb{Q}\otimes P$ is $\mathbb{Q}G$-free

I'm looking for a proof of a theorem of Swan [1, Theorem 3]:
If $G$ is a finite group and $P$ a finitely generated projective $\mathbb{Z}G$-module, then $\mathbb{Q}\otimes_\mathbb{Z}P$ is a free ...

**1**

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**1**answer

160 views

### Rank of a locally free $\mathbb Z[G]$-module

This is a basic question. Let $G$ be a finite group, $M$ a finitely generated $\mathbb Z[G]$-module so that the $\mathbb Z_p[G]$-module $M_p$ is free for all prime numbers $p$, i.e. is locally free.
...

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**0**answers

447 views

### Is this group given presentation isomorphic to $\mathbb{Z}_2$, and why? [migrated]

We have the group $G = \langle a, \,b \; | \; a^2b^2ab^{-1}, \, a^3b^4a^{-2}b^{-3}\rangle$.
Obviously $G/G' = \mathbb{Z}_2$. Is it true that $G = \mathbb{Z}_2$ or equivalently that $G'=1$? It is ...

**0**

votes

**1**answer

57 views

### Radical and Centric not Essential P-group

I'm looking that in the Fusion System categories, the p-subgroups that are essential, are centric (by definition) and radical (by implication of the definition), but I want to know if there is an ...

**5**

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**3**answers

309 views

### Character Values for Alternating Groups of degree $\geq 7$

Is it true that in each row and column of the character table of alternating groups with degree $\geq 7$ there are at most two complex values? Any reference will be highly appreciated.

**5**

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**1**answer

206 views

### Group cohomology of the cyclic group

It is well known how to compute cohomology of a finite cyclic group $C_m=\langle \sigma \rangle$, just using the periodic resolution,
$\require{AMScd}$
\begin{CD}
\cdots @>N>> \mathbb ...

**6**

votes

**2**answers

477 views

### A question about (unicity of certain cycles in a Cayley graph of a) symmetric group

Let $S=\{(1,2),(1,2,3,\ldots,n),(1,2,3,\ldots,n)^{-1}=(1,n\ldots,2)\}$ be a subset of the symmetric group $S_n$. We know that $(1,2,\ldots,n)(1,2)=(2,3,\ldots,n)$, and thus ...

**2**

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**2**answers

70 views

### p-groups as finite union of disjoint normal abelian subgroups

I was interested in knowing if groups with following property have been studied( like what can be said about structure of the group) :
"$G$ can be written as disjoint union of a given number of ...

**4**

votes

**1**answer

64 views

### Exterior Powers of finite abelian group

Let $A$ be a finite $\mathbb{Z}$-module (i.e., a finite abelian group). My question is: for what $n\in \mathbb{Z}^{n\geq 2}$ the map
\begin{align}
\alpha_{n}:\bigwedge^nA&\to A^{\otimes n}\\
...

**4**

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**1**answer

213 views

### Finite group action on quasi-projective varieties

Let $X$ be a smooth, quasi-projective variety, $G$ be a finite group which acts freely and properly on $X$. Denote by $\alpha:X \to X/G$ the quotient. Is $\alpha$ generically etale?
Also, as I am ...

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**1**answer

166 views

### Is there a characterization of CI-groups of order less than 100?

We know some benefit criterion in articles such as:
C. H. Li, On isomorphisms of finite Cayley graphs-a survey, Discrete Math., 256 (2002) 301-334.
C. H. Li, Z. P. Lu, P. ...

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**1**answer

186 views

### Is there a formula for the number of elements in $S_n$ having length $k$ with respect to the generators taken to be the transpositions?

Define the length of a permutation as the minimum number of adjacent transpositions needed to describe it.
We know that there is only one element of length $0$ in $S_n$ and $n-1$ elements of length ...

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**0**answers

90 views

### A finite supersolvable group with generators of prescribed order

Let $G=\langle a,b\rangle$ be a finite supersolvable group. Is there any special information about the structure of $G$ when $o(a)=2$ and $ o(b)=2^k > 2$?

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**6**answers

467 views

### Transitive permutation groups which all of their proper subgroups are intransitive

Let $G$ be a transitive permutation group on a finite set $\Omega$. It is clear
that if $G$ is regular, then every proper subgroup of $G$ is intransitive. Is
there any other class of groups with this ...

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**0**answers

37 views

### Gramian of a permutation group orbit

let $W\in R^{d\times k},d>k$. Suppose that the associated gramian has the following structure:
$$
W^TW=(P_{1}t,\cdots,P_{k}t)
$$
with the set $\{P_{i},i=1,\cdots,k\}$ forming a group of ...

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70 views

### An optimal lower bound related to generators in a boolean interval of finite groups

Let $[H,G]$ be a rank $n$ boolean interval of finite groups (i.e. $[H,G] \simeq B_n$ as lattice).
Let the set $E = \{ g \in G \ | \ \langle H,g \rangle = G \}$
Remark: If $g \in E$ then $Hg ...

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124 views

### Lifting automorphisms of quotient groups

I am concerning here a natural question:
Problem: Let $G$ be a finite group, and let $N$ be a characteristic subgroup of $G$. When can an automorphism $\varphi\in\mathrm{Aut}(G/N)$ be lifted to an ...

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votes

**2**answers

116 views

### Finite solvable groups with abelian Fitting subgroup

Let $G$ be any finite solvable group with Fitting subgroup
$F(G)$. Which conditions on $F(G)$ makes $G$ to be supersolvable?
(It is well-known that any finite solvable group with cyclic Fitting ...

**15**

votes

**5**answers

470 views

### Cayley graphs of $A_n.$

Consider the Cayley graphs of $A_n,$ with respect to the generating set of all $3$-cycles. Their properties must be quite well-known, but sadly not to me. For example: what is its diameter? Is it an ...

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votes

**2**answers

182 views

### Finding an “optimal” quotient in a free group

Consider the abelian free group $G = \mathbf{Z}^n$ of rank $n$ and a finite subset $A \subset G \setminus \{0\}$. Since $G$ is residually finite, there is a subgroup $H \subset G$ such that $A \cap H ...

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**3**answers

231 views

### Signs in Chevalley's commutator formula

I am trying to understand presentations of twisted groups of Lie type (specifically $^2D_5$) over finite fields using Steinberg's presentations (for instance from Gorenstein, Lyons and Solomon, Number ...

**2**

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**1**answer

102 views

### Intransitive finite irreducible linear groups whose orbits are all large

I am interested in intransitive irreducible linear subgroups $G\subseteq\mathrm{GL}_n(\mathbb{F}_p)$ acting on $V-\{0\}=\mathbb{F}_p^n-\{0\}$ in the natural way, such that all of the orbits are very ...

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**0**answers

107 views

### A question about $p$-solvable groups

$H$ is called $s$-permutable in $G$ if it permutes with every Sylow subgroup of $G$.
$H$ is called $s$-permutably embedded in $G$ if each Sylow subgroup of $H$ is a Sylow subgroup of some ...

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156 views

### Detecting a module for the free group algebra on a finite quotient

Let $F_2$ be the free group on two generators $x,y$ and let $R$ be the group algebra $\mathbf{Q}[F_2]$. Let $a,b,c$ be integers. Then define a right $R$-module
$M = R / (ax + by + c) R$.
I am ...

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183 views

### Weyl Groups as Galois groups

I am looking for explicit examples (for all positive integers $n \ge 5$) of degree $2n$ even polynomials $f(x)=h(x^2)$ over the field $\mathbb{Q}$ of rational numbers such that the Galois groups of ...

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**0**answers

188 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 ...

**6**

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**1**answer

199 views

### Are there familiar expressions for (finite) joins of finite groups?

Milnor construction of the classifying space of a topological group $G$ is given in terms of infinite joins of $G$. Schwarz then proved that the $k+1$ iterated self join of a group $G$ classifies ...

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**1**answer

117 views

### A decomposition of $w_0$ which is similar to the reduced decomposition

Some basic definitions about reduced decomposition:
In the symmetric group $S_n$, let $s_i$ denote the adjacent transposition $(i,i+1),i\in \{1,2,\cdots,n-1\}.$ Since $S_n$ is generated by adjacent ...

**5**

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**1**answer

167 views

### A question on the commutativity degree of the monoid of subsets of a finite group

The commutativity degree $d(G)$ of a finite group $G$ is defined as the ratio
$$\frac{|\{(x,y)\in G^2 | xy=yx\}|}{|G|^2}.$$It is well known that $d(G)\leq5/8$ for any finite non-abelian group $G$. If ...

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**1**answer

243 views

### Does the Fano plane “embed” in the complex projective plane?

$PSL(2,7)$ acts on the projective plane over $\mathbb{F}_2$ (the Fano plane) through its identification with $GL(3,2)$. It also acts on the projective plane over $\mathbb{C}$ through either of its ...

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104 views

### On the structure of groups according to their conjugacy classes

A finite group $G$ satisfies property $P_n$ if for every prime integer $p$, $G$ has at most $(n-1)$ non-central conjugacy classes the order of the representative element of which is a multiple of $p$. ...

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114 views

### Cocycle condition for 2-groups

I know that if $\omega_d(g_1, \ldots, g_d)$ is an d-cocycle characterized by $H^d(G,U(1))$, it satisfies the co-cycle condition
$(d\omega_d)(g_1, \ldots, g_{d+1}) = g_1.\omega_d(g_2,\ldots,g_{d+1}) + ...

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**1**answer

64 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 ...

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**1**answer

304 views

### Is $[729,57]$ a Sylow $3$-subgroup of some well-known group?

Let $G$ be the group $[729,57]$, using GAP's notation. I have so far two descriptions of the group:
a presentation
an embedding (not surjective!) of the group into a Sylow $3$-subgroup of the unit ...

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**1**answer

191 views

### Finite groups $G$ satisfying property $P_n$

A finite group $G$ satisfies property $P_n$ if for every prime integer $p$, $G$ has at most $(n−1)$ non-central conjugacy classes the order of the representative element of which is a multiple of $p$. ...

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**0**answers

71 views

### Groups with special character degrees

Let $G$ be a finite group of order $p_1^{a_1}\times p_2^{a_2}\times\cdots \times p_n^{a_n}$. Is there any classification for simple groups such that for each $i$, $p_i^{a_i}$ is an irreducible ...

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**2**answers

89 views

### sum of squares of Schur polynomials indexed over partition valued functions on a set

Fix a finite set $X$ and two natural numbers $d$ and $n$.
For a partition $\lambda$ and a number $d$ denote by $s_\lambda^d(x_1,\dots,x_d)$ the Schur polynomial in $d$-many variables $x_1,\dots,x_d$. ...

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**3**answers

446 views

### Which groups are Galois over some p-adic field?

Suppose I have some finite $p$-group $G$, or a little extension of it.
How do I know if there exists a prime $l$ and a finite extension $K$ of $\mathbb{Q}_l$ such that $G$ is the Galois group of ...

**6**

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**1**answer

530 views

### Positivity of the alternating sum of indices for boolean interval of finite groups

Let $G$ be a finite group and $H$ a subgroup such that the interval $[H,G]$ is a boolean lattice.
Let $L_1, \dots , L_n$ be the maximal subgroups of $G$ containing $H$.
Let the alternative sum ...

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**1**answer

103 views

### Can the reversed lattice of a subgroups interval be represented?

Let $G$ be a finite group and $H$ a subgroup. The interval $[H,G]$ is the lattice of overgroups of $H$.
It is an open problem to know if every finite lattice can be represented by such an interval ...