Questions on group theory which concern finite groups.

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### Subgroups of powers of the alternating group on 5 elements

Definition: Let $G$ be a group, and let $H \leq G$ be a subgroup. We say that $H$ is big in $G$ if for every intermediate subgroup $H \leq L \leq G$ there exists some $x \in L$ such that $\langle H ...

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63 views

### When does a finite group have a lower-dimensional representation than one of its quotients?

The minimal dimension of a nontrivial representation of a central extension of a group $G$ can be smaller than the minimal dimension of a nontrivial representation of $G$. What about non-central ...

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**2**answers

564 views

### divisors of $p^4+1$ of the form $kp+1$

In group theory the number of Sylow $p$-subgroups of a finite group $G$, is of the form $kp+1$.
So it is interesting to discuss about the divisors of this form. As I checked it seems that for an odd ...

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**1**answer

159 views

### The groups with symmetric subgroups lattice

Let $G$ be a group and $\mathfrak{L}(G)$ be set of all subgroups of $G$. Clearly, $\mathfrak{L} (G)$ is a lattice.
If we know that $\mathfrak{L} (G)$ is symmetric then what can be said about the ...

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**1**answer

188 views

### Is a prime index inclusion of finite groups, separating?

Let $(H \subset G)$ be an inclusion of finite groups.
Let $\{ g_i \ \vert \ i \in I=[1, \dots ,n] \}$ a subset of $G$ of double coset representatives, i.e. $$G = \coprod_{i \in I} Hg_iH$$
On the ...

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59 views

### Minimal rank of a permutation resolution of a $G$-lattice

Let $G$ be a finite group.
By a $G$-lattice I mean a finitely generated free abelian group $L$ with an action of $G$.
One says that $L$ is a permutation lattice if $L$ has a $\mathbb{Z}$-basis ...

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**1**answer

162 views

### The lower bound of a group with characters of special degrees

Is there any lower bound for the order of a group with an irreducible character of degree $p$, where $p$ is a prime.
Is there any similar result for $p^2$ or $p^3$ instead of $p$?
Thanks for your ...

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470 views

### classification of $p$-groups

I have two questions regarding to $p$-groups.
A $p$-group $G$ is said to be extraspecial of $G'=Z(G)$ has order $p$. Hence extraspecial groups are examples of $p$-groups with cyclic center. Of ...

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35 views

### Is there a better rank bound for fibered products?

Let $G$ be a profinite group of rank $d \geq 2015$, which is a fibered product of two groups of rank at most $2$. That is, there exist closed normal subgroups $N_1, N_2 \lhd_c G$ with $N_1 \cap N_2 = ...

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220 views

### A simple group that its order divide order of an alternating group

Let $G$ be a simple group such that
1) $|G|\mid|\mathrm{Alt}_{p}|$
2) $p\mid |
G|$, and $p>13$ is prime.
3) $G$ hasn't any elements of order $rp$ for every prime number $r$.
My question: ...

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90 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, ...

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**1**answer

115 views

### about subgroup of general linear group [closed]

Thanks for any comments
Let $G=GL_n(F)$ be general linear group over finite field $F$. Consider two isomorphic subgeoup $H_1,H_2$ of $G$ such that $H_i\cong GL_k(\bar{F})$, where $\bar{F}$ is an ...

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110 views

### Finite groups normalizing a torus

Let $G$ be a semi-simple linear algebraic group over the complex numbers, e.g. the special linear group. Can you find an example of a finite sub-group $H$ of $G$ which does not normalize any maximal ...

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**1**answer

100 views

### What is the corank of a proper char subgroup of a finite index subgroup of a free group?

Let $F$ be the free group of rank $\aleph_0$, let $L \leq F$ be a finite index subgroup, and let $M$ be a proper characteristic subgroup of $L$. Is it possible that there exists a finite subset $S ...

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69 views

### Orbits of a cyclic group on the powerset [closed]

Let $G$ be a finite cyclic group and denote by $\mathcal{P}(G)$ its powerset. Then $G$ acts on $\mathcal{P}(G)$ by acting on each element in a subset $S\in\mathcal{P}(G)$individually. Is there any ...

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166 views

### Generalization of the fundamental theorem of cyclic groups 2

This post is a sequel of Generalization of the fundamental theorem of cyclic groups
Let $G$ be a finite group then the fundamental theorem of cyclic groups can be formulated as follows:
Theorem: $G$ ...

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**1**answer

447 views

### Generalization of the fundamental theorem of cyclic groups

Let $G$ be a finite group then the fundamental theorem of cyclic groups can be formulated as follows:
Theorem: $G$ is cyclic iff it admits no two different subgroups with the same order.
proof: see ...

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**1**answer

170 views

### Finite solvable groups are generated by a nilpotent subgroup + K elements?

Is there a constant $K \in \mathbb{N}$ such that for every finite solvable group $G$, there exists a nilpotent subgroup $N \leq G$, and a subset $S \subseteq G$ with $|S| \leq K$, and $\langle ...

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104 views

### Does every connected vertex transitive graph on $n$ vertices (except for $C_n$) have minimum feedback vertex set of size $\Omega(n)$?

Feedback vertex set is a set of vertices whose removal leaves an acyclic graph.
It is known that every vertex transitive graph on $n$ vertices has minimum vertex cover of size $\Omega(n)$. It is also ...

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245 views

### Extensions of $SL(2,\mathbb{F}_q)$

Let $n = q(q^2-1)$ (the order of $SL(2,\mathbb{F}_q)$ I think). How many groups of order $2n$ contain $SL(2,\mathbb{F}_q)$ as a (necessarily normal) subgroup? Is this number known exactly? Seems ...

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119 views

### scalar multiple of Young symmetrizer

The following is a lemma from Fulton and Harris' book -Representation theory,a first course (page 53):
Lemma: For all $x\in \mathbb{C}\mathfrak{S}_r$, $c_{\lambda}\cdot x\cdot c_{\lambda}= scalar ...

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212 views

### Are such averages known with representations of $S_n$?

Like is there a sense in which one can quantify that for two group elements (in different conjugacy classes) their characters are "close" for some fixed irreducible representation? (feel free to ...

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272 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 $1,2,\ldots,n$ ...

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234 views

### Embedding $\mathrm{PGL}(n,q^h)$ in $\mathrm{PGL}(nh,q)$

It is not very hard to see that for each prime power $q$ and natural numbers $n,h$, we have an embedding
$$\iota \colon \mathrm{GL}(n,q^h) \hookrightarrow \mathrm{GL}(nh, q),$$
obtained by choosing a ...

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113 views

### How does one calculate/estimate/guarantee the girth of a non-Abelian Cayley graph?

This question is in reference to this other question,
Can someone point out references (or explain!) which give techniques of being able to prove for any Cayley graph this property of having a girth ...

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132 views

### How generic are Cayley graphs of non-Abelian groups with logarithmic girth?

Given a non-Abelian group $G$ I want to choose a symmetric generating set $S \subset G$ such that $Cay(G,S)$ has girth logarithmic in the size of the set. I want to know,
For which $G$ can the ...

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290 views

### Is anything known about the eigenspectrum of the regular representation of the permutation group?

I am looking for information like upper bounds on how many times any eigenvalue can occur or something like how many eigenvalues can be there in some given range. Is anything like this known?
The ...

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134 views

### Does every group that satisfies the maximal permutizer condition then satisfy the permutizer condition?

The
permutizer of a subgroup $H$ of $G$ is defined to be the subgroup generated by all cyclic subgroups of $G$ that permute with $H$,
i.e. $\langle x \in G | \langle x \rangle H = H \langle x \rangle ...

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742 views

### learning Deligne-Lusztig theory

Can someone give me a roadmap for learning Deligne-Lusztig theory? (Except for the original article by Deligne and Lusztig)
Edit: You may assume knowledge of representation theory of finite groups ...

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149 views

### Generating finite groups using subgroups

For finite groups $L \leq K$ we define $d(L,K)$ to be the least $n \in \mathbb{N}$ for which there exist $a_1, \dots, a_n \in K$ such that $\langle L, a_1 \dots, a_n \rangle = K$.
Is there some $m ...

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64 views

### Why is the polynomial relating the invariants of a binary polyhedral group fixed by an overgroup?

Let $G$ be a finite subgroup of $\mathrm{SL}(2,\mathbb{C})$ and $N \triangleleft G$ a normal subgroup. Let $x, y, z$ be the fundamental invariants for the standard action of $N$ on $\mathbb{C}^2$, ...

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430 views

### Upper bound of |Aut(G)| for a p-group

If G is a p-group which is finitely generated with order p^n then what is the upper bound of |Aut(G)|.

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236 views

### category theoretic approach to Sylow theorems and finite group theory?

Is there a category theoretic approach to Sylow theorems?
More generally, is there an exposition of finite group theory in terms of category theory which would include Sylow theorems and little facts ...

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59 views

### Is the size of maximum matching in vertex transitive 3-uniform hyper-graph on $n$ vertices always $\Omega(n)$?

What is the best known lower bound on the size of the maximum matching in a vertex transitive $3$-uniform hyper-graph?

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346 views

### A problem with pointwise stabilizer subgroups of fixed-point subspaces II

Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$.
Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$.
Let ...

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103 views

### A problem with pointwise stabilizer subgroups of fixed-point subspaces I

Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$.
Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$.
Let ...

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**1**answer

255 views

### Is there a nonabelian finite simple group with Grothendieck ring of multiplicity one?

Let $G$ be a finite group. It admits finitely many irreducible complex representations $H_1, \dots, H_r$ which generate, for $\oplus$ and $\otimes$, the Grothendieck ring $\mathcal{G}(G)$ of $G$ (also ...

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125 views

### Unipotent orbit in adjoint group over finite field

[Editted: The assertion is wrong; see Jay's answer]
My apology if this question is too simple. I am reading Deligne-Lusztig "Reductive groups over finite fields" and at the beginning of Chap. 4, ...

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212 views

### Matrix Elements of Real Representations

I asked this question over at Math.StackExchange and despite having had a bounty on it I did not receive an answer.
Suppose that $G$ is a finite group and we have a unitary irreducible representation ...

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**1**answer

166 views

### Finite p-groups and their fibered products

Is every finite $p$-group an epimorphic image of a fibered product of two finite $p$-groups which can be generated by $2$ elements?

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486 views

### Enumeration of $0-1$ matrices with determinant $1$

Has the number $f(n)$ of $n \times n$, $0{-}1$ matrices whose determinant is $+1$
been enumerated?
E.g., for $n{=}2$, there are $f(2)=3$ such matrices:
$$
\left(
\begin{array}{cc}
1 & 0 \\
0 ...

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307 views

### Maximal abelian subgroup of general linear groups

Thanks for any help or comments.
Is it possible to recognize all maximal abelian subgroups of general linear group on finite field $F$ of order $q$, $GL_n(F)$.
By maximal abelian I mean if $A$ is ...

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139 views

### Subgroups of index 2 in a fibered product

Let $G$ be a finite group and let $M,N \lhd G$ be normal subgroups with a trivial intersection. Suppose that $G$ has a subgroup of index $2$. Must $G$ have a subgroup of index $2$ which contains ...

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144 views

### Invariant subspaces of an $F_2$-representation of the affine linear group of dimension 1

Let $p$ be an odd prime (large if it matters) and let $G= Aff(\mathbb{F}_{p^2}) \cong \mathbb{F}_{p^2} \rtimes \mathbb{F}_{p^2}^*$ be the affine linear group acting on $\mathbb{F}_{p^2}$ by $x\mapsto ...

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### How do you *state* the Classification of finite simple groups?

From the point of view of formal math, what would constitute an appropriate statement of the classification of finite simple groups? As I understand it, the classification enumerates 18 infinite ...

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173 views

### About the set of Sylow-$p$ subgroups of $G$

Let G be a finite group and S be the set of Sylow p-subgroups of G for a
prime p dividing the order of G. Assume that |S|>1.
Let U and V be two disjoint non-empty subsets of S such that,
...

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64 views

### Experimenting with the spider relator

The Monster group (actually the bimonster) has a presentation as Y555. Y555 is the quotient of a coxeter group (the coxeter diagram is a central node with three "spokes" coming out of it with length ...

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79 views

### finite p-group subgroup of infinite p-group

is there any finite p-group G that is subgroup or minimal/maximal subgroup of infinite p-group H?
if yes what is the limits? can this happen with different p's?
i'm more interested in being maximal ...

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173 views

### Deligne-Lusztig and Character sheaves

Consider: $G$ - a nice group ($GL_n$) over a finite field $F$. $X$ - the flag variety. Consider a nice $G$-equivariant $l$-adic sheaf $\mathcal{M}$ on $X \times X$, equipped with Weil structure.
Fix ...

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210 views

### permutation action on cohomology of Stiefel manifolds

Let $V_k(\mathbb{R}^n)$ be Stiefel manifolds.
In the paper The cohomology rings of real Stiefel manifolds with integer coefficients, Martin Čadek, Mamoru Mimura, and Jiří Vanžura, J. Math. Kyoto ...