Questions about the branch of abstract algebra that deals with groups.

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13
votes
2answers
649 views

Torsion-freeness of two groups with 2 generators and 3 relators and Kaplansky Zero Divisor Conjecture

Let $G_1$ and $G_2$ be the groups with the following presentations: $$G_1=\langle a,b \;|\; (ab)^2=a^{-1}ba^{-1}, (a^{-1}ba^{-1})^2=b^{-2}a, (ba^{-1})^2=a^{-2}b^2 \rangle,$$ $$G_2=\langle a,b \;|\; ...
1
vote
0answers
133 views

the braid groups on the sphere

The braid group $B_n(S^2)$ on the sphere possesses a finite element of order which generates a cyclic group $M$ in $B_n(S^2)$. My question is this: What is the index of $H$ in $B_n (S^2)$? Same ...
2
votes
0answers
79 views

Dicks–Dunwoody almost stability theorem

In the book 'Groups acting on graphs' (1989), Dicks and Dunwoody prove the following theorem (paraphrased): Let $G$ be a group acting on a set $E$, let $E'$ be a subset of $E$ and let $V$ be the set ...
1
vote
1answer
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 ...
3
votes
0answers
57 views

Infinite finitely generated groups whose Frattini factors are Klein 4-group

Is there an infinite finitely generated group whose Frattini factor is isomorphic to Klein 4-group?
4
votes
1answer
308 views

Maximal set of non commuting elements in a conjugacy class of $S_8$

Can someone calculate by computer or prove the subset of maximal order of pairwise non commuting elements in the set of conjugacy class containing $(123)(45)$ in $S_8$? I mean a subset of conjugacy ...
-1
votes
1answer
115 views

Can someone explain how Sims's algorithm works on a permutation group with a simple example? [closed]

Can someone explain how Sims's algorithm works on a permutation group with a simple example? The book "Permutation group algorithms" by Seress is a pretty hard read with a whole bunch of confusing ...
2
votes
1answer
242 views

A presentation for $GL(2,\mathbb{Z}/p^n \mathbb{Z})$

In 'A presentation of $PGL(2,p)$ with three defining relations' by E.F.Robertson and P.D.Williams, we can find a presentation of $PGL(2,p)$: $\langle a,b | a^2 = b^p = (a b^2 a b^r)^2 = (abab^r)^3 = ...
0
votes
1answer
81 views

Uniform pro-p groups as a semi-direct product

Let $G$ a finitely generated uniform pro-$p$ group. Then $G/[G,G]$ is abelian and so it is of the form $\mathbb{Z}_p^r\times T$ for some integer $r$ and finite $p$-group $T$. Therefore, $[G,G]$ is ...
3
votes
3answers
660 views

Square of non-zero element in group algebra is always non-zero?

Consider a finite group $G$ and complex group algebra $\mathbb{C}(G)$, i.e. formal sums $$ \sum_{g \in G} a_gg, \ a_g \in \mathbb{C},$$ with algebra structure: $$ \sum_g ...
5
votes
1answer
233 views

Zero-sum sets in union-closed families

The Davenport constant $D(G)$ of a finite abelian group $G$ is the minimum integer $n$ such that whenever $a_1, \ldots, a_n \in G$ (not necessarily distinct), there is a non-empty $I \subseteq [n]$ ...
4
votes
1answer
110 views

boundary of semihyperbolic groups

There are various definitions of boundary of a hyperbolic group. Which of those generalize to semi-hyperbolic groups (in the sense of Alonso and Bridson)? The example I have in mind is a semisimple ...
6
votes
1answer
121 views

Class number of Burnside groups

Let $B(m,n)$ be the Burnside group on $m$ generators of exponent $n$. Suppose the class number - the number of conjugacy classes - of $B(m,n)$ is finite. Does it imply that $B(m,n)$ is finite?
1
vote
1answer
99 views

Centralizer of a central involution in a simple group of Lie type

Let $G$ be a finite simple group of Lie type and $x$ be a central involution (that is, an involution which is contained in the center of a Sylow $2$-subgroup). Is it true that, if $y$ is another ...
1
vote
1answer
113 views

When the reduced $C^*$-algebra of $\Gamma$ admits character then $\Gamma$ is amenable [closed]

Suppose that $C^*_r(\Gamma)$ admits some character (homomorphism into $\mathbb{C}$)-here $\Gamma$ is discrete group and $C^*_r(\Gamma)$ is the closure of the image of the group ring $\mathbb{C}\Gamma$ ...
1
vote
0answers
88 views

to what extent is a reductive group hyperbolic?

The group $SL(2,F)$ where $F$ is a local nonArchemidian field is hyperbolic. Various generalizations of the notion of hyperbolicity have been studied in the literature (I've seen terms like ...
6
votes
3answers
323 views

Asymptotics for the number of abelian groups of order at most $x.$

The number of abelian groups of order $n$ (call it $a(n)$ is a studied subject (see http://oeis.org/A000688), but I can't seem to find any asymptotic results. Obviously, there is no asymptotic for ...
0
votes
0answers
56 views

HNN extension group with finitely generated base

Let $B$ be a group and let $A_1$ and $A_2$ subgroups of $B$ with $\phi :A_1\rightarrow A_2$ an isomorphism. Let $\left<t\right>$ be the infinite cyclic group, generated by a new element $t$. The ...
2
votes
1answer
313 views

One-dimension Algebraic groups

I am searching for a possible analogue of a result in algebraic groups in a non-commutative setting, so I am looking for different proofs of the following : Let $K$ be an algebraically closed field. ...
1
vote
1answer
125 views

p-groups with unique normal minimal subgroup

Have $p$-groups with a unique normal minimal subgroup been classified? Is there any article on the subject?
8
votes
1answer
140 views

Schur multiplier of $Sp(2g, \mathbb{Z}_2)$ for $g \geq 3$

This question is about the computation of $H_2(Sp(2g, \mathbb{Z}_2), \mathbb{Z})$, where $Sp(2g, \mathbb{Z}_2)$ is the group of symplectic $2g \times 2g$ matrices over $\mathbb{Z}_2$. With respect to ...
0
votes
1answer
67 views

About $c(A)$ in $c(A)|A|\leq |A^{-1}A|$

Let $G$ be a finite group, $\emptyset\neq A\subseteq G$, $A^{-1}:=\{ a^{-1}:a\in A\}$, and put $$c(A):=\max\{t\in \mathbb{Z}: t|A|\leq |A^{-1}A|\}$$ It is clear that $1\leq c(A)\leq ...
2
votes
0answers
150 views

Groups with isomorphic quotients [closed]

Assume we have a finitely presented group $G$ and a non-trivial normal subgroup N. How can one decide that $G/N$ is isomorphic to $G$ or not? $G$ is given as a presentation and $N$ as a set of words.
2
votes
0answers
97 views

Computing abelianizations of some explict finite subgroups of $GL_2(\mathbb{Z}/p^n \mathbb{Z})$

I have been attempting to find some abelianizations of some subgroups of $GL_2(\mathbb{Z}/p^n \mathbb{Z})$. I have been using brute force for the most part but I get very messy results. Here is an ...
3
votes
2answers
99 views

References about the matrix generators of the finite subgroups of the orthogonal group O(4)

"On Quaternions and Octonions" by Conway and Smith gives the classification of the finite subgroups of the orthogonal group O(4). I want to get the explicit matrix generators of the finite subgroups. ...
8
votes
1answer
163 views

Nonsolvable finite quotients of matrix groups

Suppose that $\Gamma$ is a finitely generated nonsolvable subgroup of $GL(n, R)$. Is it in the literature that $\Gamma$ has a nonsolvable finite quotient? I know how to prove it (the hardest ...
1
vote
0answers
138 views

Braids with an infinite number of strings

Has anyone developed a theory for braids with an infinite number of strings?
1
vote
1answer
147 views

Is there a nice choice-free argument to count the number of sublattices?

It's a well known fact that the number of index $n$ sublattices of a rank two lattice $\Lambda$ is given by $\sigma_1(n) = \sum_{d\mid n} d$. Here is a proof of this fact: Proof: choosing a basis of ...
-3
votes
1answer
133 views

Even-odd partitioned groups! [closed]

Let $G$ be a group with the property $G=G_e\dot{\cup} G_o$ with $G_oG_o\subseteq G_e\leq G$. ($\dot{\cup}$ denotes disjoint union, $\leq$ is subgroup notation, and $G_o^{-1}=\{x^{-1}: x\in G_o\}$) ...
6
votes
2answers
202 views

Uniqueness of the fusion ring for simple finite group

We know that the irreducible representations $R_i$ of a group $G$ can give rise to a fusion ring: $R_i\otimes R_j = \oplus_k N^{ij}_k R_k$. I wonder if the following statement is true or not: If $G$ ...
8
votes
1answer
185 views

Finite groups: equations with many solutions

Let $\omega$ be a word in the free group on generators $x_1,x_2,\ldots,x_n,g_1,g_2,\ldots,g_k$, where $n>0$ and $k\geq 0$. For any finite group $G$ and elements $g_1,g_2,\ldots g_k$ in $G$ we ...
3
votes
1answer
169 views

finitely presented subgroup and free solvable group of class 3

Let $F(n)$ be free group of rank $n\geq 2$. Denote by $F_d(n)$ the d-th derived subgroup, that is $F_d(n)=[F_{d-1}(n),F_{d-1}(n)]$ where $F_0(n)=F(n)$. The free solvable group of rank $n$ and ...
4
votes
0answers
143 views

Iwasawa theory, $\mathbb{Z}_p^{2}$-extension, Greenberg module

Take $H\subset \bar{\mathbb{Q}}$ be a quartic imaginary number field such that $\operatorname{Gal}(H/\mathbb{Q})=\mathbb{Z}_2 \times \mathbb{Z}_2$. Denote by $F$ the quadratic real subfield of $H$ and ...
4
votes
2answers
126 views

$2$-cohomology group of semi-direct products

Let $G=N\rtimes T$ and let $A$ be a $G$-module with a trivial $G$-action. The action of $T$ on induce a natural action of $T$ on the second cohomology group of $N$. Denote by $H^2(N,A)^T$ the ...
17
votes
2answers
650 views

Solving equations in SO(3) : an open problem by Jan Mycielski

I am interested in a problem closely related to a problem stated by Jan Mycielski in his paper Can One Solve Equations in Group? (The American Mathematical Monthly, 1977, ...
21
votes
4answers
903 views

Technical issue in the approach to Lie groups taken in a book

I'm teaching Lie groups and Lie Algebras out of Brian C. Hall's book (Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer), which I've enjoyed using. I'm confused about ...
12
votes
1answer
142 views

An inequality for the minimal number of generators of a finite group

Let $G$ be a finite group, $n(G)$ the minimal number of generators and $m(G)$ the minimal number of irreducible complex representations generating (with $\otimes$ and $\oplus$) the left regular ...
5
votes
2answers
198 views

Describing Levi factors and unipotent radicals of parabolic subgroups in classical groups

I asked this question before at Math.SE (link) but got no answer. Let $G$ be an algebraic group over an algebraically closed field $k$ of characteristic $p \geq 0$. Then any parabolic subgroup $P$ ...
2
votes
0answers
133 views

Groups with Abelian Automorphism Group

In a paper, the authors Jonah-Konvisser say Until recently (~1975), there were no published examples of non-abelian groups with abelian automorphism groups. Heinken and Liebeck have methods for ...
2
votes
1answer
386 views

A group algebra isomorphism problem

What are the groups $G$ and fields $\Bbb K$ for which $\Bbb K[G]\cong\Bbb K^{|G|}$ holds? For example $\Bbb R[\Bbb F_2^n]\cong\Bbb R^{2^n}$ holds.
3
votes
0answers
145 views

Are all finitely generated subgroups of SL2 LERF? [closed]

Let $H \leq \mathrm{SL}_2(\mathbb{R})$ be a finitely generated subgroup. Must $H$ be LERF? A group $H$ is said to be LERF (locally extended residually finite), or subgroup separable, if its ...
2
votes
1answer
270 views

Groupoid isomorphism vs. group isomorphism

Assume that $\Gamma$ is a group with neutral element $e$. We associate to $\Gamma$ the following groupoid $G$: $G=\Gamma \times \Gamma,\;\;\;G^{(0)}=\Gamma \times \{0\},\;\;s(a,b)=(a,e),\;\;\; ...
0
votes
0answers
68 views

Expected shortest word length\depth in Braid groups from set of all braids with length L

Consider the set of all positive braids on n strands, with a fixed length L. $$B^+_{n,L}:=\{\beta\in B_n:\beta=\sigma_{i_1}\sigma_{i_2}\ldots\sigma_{i_L},1\leq i_k \leq n-1\}$$. Using the relations, ...
2
votes
0answers
166 views

When can we establish an isomorphism between two not-finitely presented groups?

Assume that finitely generated groups $G$ and $H$, are not finitely presented. Fix a generating set $\mathfrak g:=\{g_1,\dotsc,g_n\}$ of $G$. Let $\mathfrak R:=\{R_1,R_2,\dots\}$ be the set of all ...
2
votes
1answer
142 views

Counting Frobenius groups with abelian Frobenius complement

In R. Brown, D. K. Harrison Abelian Frobenius kernels and modules over number rings. J. Pure Appl. Algebra 126 (1998), no. 1-3, 51–86, Remark 11.13 (A), the authors show that the number of isomorphism ...
3
votes
1answer
155 views

Must normalizing field outer automorphisms “divide” the dimension?

Imprecise question: To get a normalizing field outer automorphism of order $r$, must we multiply the dimension by $r$? Precise hypothesis: Let $p\geqslant 5$ be a prime, let $q$ be a power of $p$ and ...
4
votes
1answer
196 views

The parity of the full automorphism group order of finite non-abelian groups of prime exponent

Is there a finite non-abelian group $G$ of prime exponent such that the full automorphism group of $G$ is of odd order?
9
votes
2answers
342 views

Characters of permutation groups

Let $N$ be a fixed positive integer, and denote by $C(m)$ the number of permutations on an $N$-element set that have exactly $m$ cycles (counting $1$-cycles). Then it is in the literature that the ...
4
votes
1answer
258 views

What is known about the algebraic variety defined by the group determinant?

What is known about the algebraic variety $V_G$ defined by $det(X_G) = 1$ where $X_G$ is the group matrix $(x_{g_ig_j^-1})$ of a finite group $G$? It is known that two finite groups having the same ...
2
votes
1answer
129 views

Could the number of defining relators of a finitely presented group increase

Do there exist finitely generated groups $G$ and $H$ with following properies: $G=\langle g_1,\dotsc,g_n\rangle$ is not finitely presented, Let $R:=\{r_1,r_2,\dots\}$ be the set of its defining ...