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6
votes
4answers
376 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 ...
2
votes
1answer
107 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, ...
3
votes
1answer
131 views

Is there a way to find an efficient set of relations for presenting the subgroup generated by two matrices in $SL(2, q)$?

Given two elements $a, b \in SL(2, \mathbb{F}_q)$, is there a way to find an efficient presentation $$\langle x, y \mid \text{relations}\rangle$$ of the subgroup $\langle a, b \rangle$? My intention ...
2
votes
2answers
122 views

What are the 2-generated subgroups of the special linear group $SL(2, q)$ over a finite field?

What is the subgroup structure of the subgroups $\langle a, b\rangle$ where $a, b \in SL(2, q)$?
6
votes
1answer
201 views

Is every nonabelian finite simple group a quotient of a triangle group $(a,b,c)$ with $a,b,c$ coprime?

Here by triangle group $(a,b,c)$ I mean the group with presentation $$\langle x,y \;|\; x^a = y^b = (xy)^c = 1\rangle$$ In other words, for every finite simple nonabelian group $G$, do there exist ...
4
votes
1answer
445 views

Is there a structure theorem or group law for finite groups generated by two elements?

Say that $a, b \in G$ are two elements of a finite group $G$. Is there a structure theorem for the structure of $\langle a,b\rangle$? Is there a way to derive group laws for the group operation in the ...
3
votes
1answer
164 views

A representation of a finite group where every nonzero vector has a trivial stabilizer [duplicate]

What are the finite groups which admit a non-zero representation in char 0 where every non-zero vector has stabilizer equal to $\left<1\right>$? Cyclic groups of prime order is one obvious ...
4
votes
0answers
169 views

A class 3 group of order 243

Let G be a group of order $243=3^5$. We denote by $(G_i)$ its lower central series and assume that $G$ has class $3$ and that $|G:G_2|=|G_3|=9$. We assume moreover that the cubing map factors as a ...
48
votes
1answer
3k views

Why can't a nonabelian group be 75% abelian?

This question asks for intuition, not a proof. An earlier question, Measures of non-abelian-ness was thoroughly answered by Arturo Magidin. A paper by Gustafson1 proves that, for a nonabelian group, ...
0
votes
0answers
23 views

Is there a non-solvable integral fusion category of square-free dimension?

A finite group of square-free order is solvable (see here). You can find the definition for a solvable fusion category in this paper. Question: Is there a non-solvable integral fusion category of ...
4
votes
0answers
428 views

Is a non-trivial finite perfect group of order 4n? [migrated]

A finite group $G$ is perfect if $G = G^{(1)} := \langle [G,G] \rangle$, or equivalently, if any $1$-dimensional complex representation is trivial. Question: Is a non-trivial finite perfect group of ...
2
votes
0answers
191 views

Does knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ sometimes offer a significant advantage in finding $r$?

Is there a cyclic group $\mathcal G$ with generator $g$ for which the discrete log problem is assumed to be hard, but knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ for random $r$ makes finding $r$ easy? ...
1
vote
1answer
56 views

Isotropic subspaces in a symplectic vectorspace over $GF(q)$

Let $V$ be a symplectic vectorspace of dimension $2n$, and $r\mid n$. Is this statement true?"There is an isotropic spread of $r$ dimensional subspaces in $V$". By an isotropis subspace I mean a ...
1
vote
2answers
230 views

Subgroup of Projective general linear group on complete discrete valuation ring

Let $R$ be a complete dvr and $k$ its residue field of positive characteristic. Let $H$ be a finite subgroup of $PGL_2(k)$ such that the order of $H$ is prime with $char(k)$. Is there some ...
2
votes
1answer
121 views

Asymptotic of min(#generators times diameter), for a Cayley graph of Sn

This post is a sequel of Diameter of symmetric group. Let $\Sigma$ a generating subset of $S_n$, $\Gamma(S_n, \Sigma)$ the Cayley graph and $d_{\Sigma}$ the diameter of $\Gamma(S_n, \Sigma)$. Let ...
2
votes
2answers
164 views

Permutation covering of a $G$-lattice

Let $G$ be a finite group. By a $G$-lattice we mean a finitely generated free abelian group $L$ with an action of $G$. We say that $L$ is a permutation $G$-lattice if $L$ has a ${{\mathbf{Z}}}$-basis ...
3
votes
0answers
245 views

On Thompson conjecture

Let $N(G)$ be the set of conjugacy class sizes of finite group $G$. Let $G$ be a finite group for which $N(G)=N(A_n)$, where $A_n$ is the alternating group of degree $n$. Let $p$ and $q$ be distinct ...
22
votes
0answers
406 views

Next steps on formal proof of classification of finite simple groups

While people are steaming ahead on finessing the proof of the classification of finite simple groups (CFSG), we have a formal proof in Coq of one of the first major components: the Feit-Thompson ...
0
votes
0answers
124 views

Representation of finite group

Let $\Gamma$ be a finite index subgroup of $SL_2(\mathbb{Z})$. Let $\pi$ be a natural covering from $\Gamma\backslash\mathbb{H}$ to $SL_2(\mathbb{Z})\backslash\mathbb{H}$. Denote by $\text{Deck}(\pi)$ ...
2
votes
2answers
133 views

Finite groups which have elements $g$ of order $pq$ such that the sizes of the conjugacy classes of $g^p$, of $g^q$ and of $g^{q-p}$ coincide [closed]

Let $G$ be a finite group and let $g \in G$ be an element of order $pq$, where $p < q$ are prime numbers. Denote by $g^G$ the conjugacy class of $g$ in $G$. Under which conditions does the ...
1
vote
0answers
163 views

Are the finite groups inclusions, almost all relatively cyclic?

Definition: An inclusion of finite groups $(A \subset B)$ is relatively cyclic if $\exists b \in B$ such that $\langle A,b \rangle = B$. Definition: Two inclusions of finite groups are equivalent, ...
1
vote
1answer
125 views

Groups with many vanishing elements

It is well-known that every non-linear character $\chi$ of a finite group $G$ vanishes on some elements of $G\setminus Z(\chi)$. The question is What can be said about a finite group $G$ for which ...
13
votes
2answers
430 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 ...
1
vote
0answers
114 views

Finite quotients of an infinite product of finite groups

Let $G$ be a finite group. Consider the direct product $\Gamma=\prod_{i=1}^{\infty}G$ of (countably) infinitely many copies of $G$. For every finite set of numbers $\{i_1,\ldots,i_n\}$ we have the ...
17
votes
2answers
532 views

Is there a big solvable subgroup in every finite group?

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 ...
0
votes
0answers
98 views

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 ...
2
votes
0answers
62 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 ...
17
votes
2answers
519 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 ...
4
votes
1answer
146 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 ...
1
vote
1answer
186 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 ...
4
votes
0answers
57 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 ...
2
votes
1answer
150 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 ...
3
votes
6answers
423 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 ...
0
votes
0answers
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 = ...
-1
votes
1answer
214 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: ...
3
votes
0answers
85 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, ...
1
vote
1answer
102 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 ...
1
vote
1answer
105 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 ...
2
votes
1answer
95 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 ...
1
vote
0answers
63 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 ...
3
votes
0answers
161 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: $G$ ...
5
votes
1answer
331 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 ...
4
votes
1answer
161 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 ...
1
vote
1answer
93 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 ...
5
votes
2answers
239 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 ...
3
votes
0answers
98 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 ...
0
votes
1answer
207 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 ...
21
votes
0answers
264 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 $1,2,\ldots,n$ ...
9
votes
2answers
223 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 ...
2
votes
2answers
98 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 ...