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

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3
votes
0answers
251 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
435 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 ...
1
vote
1answer
189 views

Abelianization of limit groups

Let $G_1$ and $G_2$ be limit groups, and let $C_1$ and $C_2$ be cyclic subgroups of $G_1$ and $G_2$, respectively. Question: If $G$ is the amalgamated product of $G_1$ and $G_2$ with amalgamated ...
2
votes
2answers
448 views

What is the $\mathbb Z/2$-cohomology of $\mathrm B^n(\mathbb Z/2)$?

I would like to know the cohomology groups $\mathrm H^\bullet(\mathrm B^n(\mathbb Z/2);\mathbb Z/2)$. I assume that this is a standard computation, but I'm not sure where to look up the answer (and, ...
1
vote
0answers
62 views

A discrete presentation for a free prop-$p$-group

Let $n\geq 1$ be an integer and $p$ a prime. Suppose that $\mathcal{F}(n,p)$ is the free prop-p-group of rank $n$. Question: For each pair $(n,p)$, is it known a discrete free group $\mathfrak{F}$ ...
0
votes
0answers
126 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 $\text{Deck}(\pi)$ ...
0
votes
2answers
236 views

Representation Theory of $U(N)$

(1) Is it true that the category of representations of $U(n)$ is equivalent to the category of representations of $SU(N) \times U(1)$? If so, how is it proved, or what is a good reference. (I guess ...
1
vote
1answer
98 views

Anyone have missing reference list - Kerber “Representations . . . I”

My copy of Kerber's Representations of Permutation Groups I is missing the pages containing the references. Anybody got a copy that shows such?
0
votes
0answers
164 views

Can ugly groups have derived length 3?

Definitions: All groups referred to are finite solvable. Call such a group good if it can be constructed from the trivial group using central extensions and split extensions, call a group bad if it ...
3
votes
0answers
158 views

pro-p dense subgroup in the free group

Let $G_p$ be pseudovariety of all finite $p$-group. The pro-$p$ topology on a group $G$ is the unique group topology such that the set of normal subgroups $N$ with $G/N$ in $G_p$ is a fundamental ...
3
votes
1answer
140 views

A possible presentation with 2 generators and 2 relators for $C_4 \cdot D_8$

Is there a presentation with two generators and two relators for the group $C_4 \cdot D_8$? This group is of order 32 and its IdSmallGroup in GAP is [32,15]. Also it has the following presentation ...
3
votes
1answer
250 views

Classes of finite resoluble groups which are (faithfully) representable by triangular matrices?

Let $G$ be a group, $k$ a field and $T(n,k)\subset Gl(n,k)$, the group of invertible upper triangular $n\times n$ matrices. I know that if $\rho : G\rightarrow T(n,k)$ is faithful (i.e. into) then ...
2
votes
2answers
137 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 ...
9
votes
1answer
247 views

Countable group with uncountable number of subgroups $< 2^{\aleph_0}$

Is it consistent that there is a countable group $G$ such that the cardinality of the set of subgroups of $G$ is uncountable, but strictly less than $2^{\aleph_0}$?
2
votes
0answers
119 views

The defining relations for a subgroup of $SL(2,Z)$

$SL(2,Z)$ generated by $T=\begin{pmatrix} 1 & 1\\ 0& 1\\ \end{pmatrix}$ and $S=\begin{pmatrix} 0 & 1\\ -1& 0\\ \end{pmatrix}$ has the following defining relations $S^2=(S T)^3=C,\ ...
2
votes
0answers
98 views

Minimal number of defining relators of a finite $p$-group on a minimal generating set

What is the state-of-art of the following question? Let $p$ be any prime number. For any finite $p$-group $G$, let $r_G$ denote the minimum number of defining relators in all presentations of $G$ ...
3
votes
1answer
258 views

simultaneous action of GL(n) on the matrices

Consider the action of GL(n,k) on the set MxM where M is the set of all n-by-n matrices over k given by $g.(h,l) \mapsto (ghg^{-1}, glg^t)$. Individually these actions are well studied and good ...
3
votes
1answer
140 views

minimal polynomial of unipotents in orthogonal group

Consider split orthogonal group O(2l) over a field of characteristic zero. We may assume the matrix of bilinear form to be $\begin{pmatrix} O&I\\ I&0\end{pmatrix}$. Let u be a unipotent in ...
40
votes
1answer
992 views

Transitivity on $\mathbb{N}_0$ — a 42 problem

Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition $\tau_{r_1(m_1),r_2(m_2)}$ be the ...
2
votes
0answers
93 views

pro-p topology on a free group

James Howie in the paper "The p-adic topology on a free group:a counterexample" showed that in the free group $F$ generated by $x$ and $y$,if $a=xy^2$, $b_1=x^{-2}y^{-3}$ and $b_2=x^{-2}(xy)^5$, then ...
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 ...
3
votes
1answer
164 views

Groups with $G^n \cong G$ for some integer $n$ [duplicate]

Which integers $n>2$ have the following property? There is a group $G$ such that $G^n \cong G$; and for all integers $k$ with $1<k<n$ we have $G^k\not \cong G$.
7
votes
2answers
519 views

Avoiding countable subgroups of general uncountable groups

The following problem is a general form of another problem (motivation is available there). Initially, the problems were posted together, but the first one is solved below, a solution that does not ...
13
votes
2answers
449 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 ...
2
votes
1answer
186 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 ...
0
votes
0answers
68 views

Maximal subgroups which are not open in pro-2 groups

Is it true that a pro-2 group $G$ with $G/\Phi(G)\cong C_2\times C_2$ may have a maximal subgroup which is not open? Motivation: The Frattini subgroup of a profinite group by definition, is the ...
17
votes
2answers
538 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 ...
-2
votes
1answer
85 views

Can every non-discrete topological group G be algebraically generated by a nowhere dense subset?

Is there somone help me to show that if this problem have positive Answer : Problem :Can every non-discrete topological group G be algebraically gen- erated by a nowhere dense subset ? Thank ...
0
votes
0answers
99 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 ...
5
votes
0answers
112 views

Furtwangler's Principal ideal theorem in number fields

Does anyone know a simple proof, using cohomological method of the fact that the verlagerung from a finite group G. to its commutator subgroup G', i.e. $$G/G'->(G')^{ab}$$ vanishes? The simplest ...
2
votes
1answer
150 views

Sum of irreducible complex character degrees for alternating groups

What is sum of degrees of the irreducible complex characters of the alternating groups? The background of this question is to calculate the diminsion of a maximal torus of the associated Lie algebra ...
4
votes
1answer
260 views

Irreducibility of the tensor product of two finite-dimensional irreducible group representations

Let $k$ be an algebraically closed field of characteristic 0, let $G$ be any group and $N\unlhd G$ a normal subgroup. Let $U$ be a finite-dimensional and irreducible $kG$-module, such that $U$ is also ...
6
votes
1answer
252 views

A finitely presented group with two simple relations

Is the group $G$ with the presentation $\langle x,y \;|\; x^7=1, y^2 x y=x^4\rangle$ solvable? infinite? I have computed by GAP the following fators of the derived series of $G$: $G/G'\cong C_3 ...
0
votes
0answers
21 views

Restrictions of potential tensor fields to toric subgroups

Let $G$ be a compact connected nonabelian Lie group and let $f$ be a symmetric tensor field of order $m\geq1$ on $G$. Let $T\subset G$ be a translate of a torus subgroup of $G$ with $\dim(T)\geq1$. ...
17
votes
2answers
529 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
191 views

Do discrete groups with property $(T)$ have “modest” subgroup growth?

I saw it conjectured at http://www.mathunion.org/ICM/ICM1994.1/Main/icm1994.1.0309.0317.ocr.pdf that "discrete subgroups with property $(T)$ may have modest subgroup growth." (Page 5, directly above ...
1
vote
1answer
131 views

Zero divisors of the form $1+x+y$ in the rational group algebra

Is there a finite non-ablelian group $G$ generated by $x$ and $y$ such that $1+x+y$ is a zero divisor in the rational group algbera $\mathbb{Q}[G]$ and also $x^2$, $y^2$ and $(x^{-1} y)^2$ are all ...
4
votes
1answer
150 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
2answers
171 views

What are some examples of non-commutative $\mathbb{Q}$-monoids and/or $\mathbb{R}$-monoids?

Definition 0. Let $R$ denote a commutative semiring with $0$ and $1$. By an $R$-monoid, I mean a monoid $M$ equipped with an action $R \times M \rightarrow M$ denoted $r,m \mapsto m^r$, satisfying 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 ...
5
votes
2answers
162 views

Extension splitting over Sylow subgroups

Consider an extension of groups $$(E)\ :\ 1\to N\to G\stackrel{\pi}{\longrightarrow} Q\to 1$$ and assume $N$ is abelian, $Q$ is finite, for any Sylow subgroup $S_p$ of $Q$, the pullback $(E_p)\ :\ ...
3
votes
0answers
116 views

when are local quasigeodesics global in CAT(0)

It is well-known (and easily shown) that a local quasi-geodesic (for some value of "local") in a $\delta$-hyperbolic space is global (one can compute the constants, as well, from local data). This is ...
3
votes
0answers
80 views

Is there a name for the operation that stretches out an invertible series by a factor of $m$?

The question is whether there is an established word for the transformation that starts with an invertible formal power series over a field $k$, $u(x)=xg(x)=x(1+a_1x+a_2x^2+\cdots)$ and delivers the ...
50
votes
16answers
5k views

Solving algebraic problems with topology

Often, topologists reduce a problem which is - in some sense - of geometric nature, into an algebraic question that is then (partiallly) solved to give back some understanding of the original problem. ...
0
votes
0answers
21 views

Decomposition of axial vector and vector representions of C$_{4v}$ group

Let $R$ be the orthogonal matrix corresponding to an operation in $O(3)$. If R is a proper rotation, then both vectors $\vec{V}$ and axial vectors $\vec{A}$ are transformed in the same way, $$ ...
0
votes
2answers
124 views

Is the Frattini subgroup of a f.g virtually pro-p group open?

Let $G$ be a finitely generated profinite group, and $p$ a prime number. Suppose that there exists some open pro-$p$ subgroup $H \leq_o G$. Must $G$ have only finitely many maximal open subgroups? ...
2
votes
1answer
152 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
2answers
123 views

u.p (unique product) group which is not Right ordered ($RO$)

I am looking for an example of a u.p (unique product) group which is not Right ordered. Almost any group I pick up (obviously torsion free, as u.p. group cannot have torsion element, so no use ...
8
votes
1answer
115 views

Computing a transversal of a subgroup $H$ of $G$ in expected $O(|G : H|^2 \log |G : H| + |H|)$ time

I have the book "Handbook of Computational Group Theory", by Derek Holt, and in it is a section on finding the transversal of a subgroup. Recall a transversal of a subgroup $H$ of $G$ is a single ...