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

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4
votes
1answer
252 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
245 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
19 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
500 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
178 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
123 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
143 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
159 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
184 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
161 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
112 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
75 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 ...
0
votes
0answers
53 views

Quotient of the Baumslag-Solitar group $BS(1,m)=\langle a,b| bab^{-1}=a^m\rangle$ [migrated]

I would like to prove that in any non-trivial quotient of the Baumslag-Solitar group $BS(1,m)$ defined by, $$BS(1,m)=\langle a,b| bab^{-1}=a^m\rangle$$ the image of one of the generators $a$ or $b$ ...
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
17 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
116 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
149 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
103 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
99 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 ...
10
votes
0answers
285 views

On sentences true in all finite groups (revisited)

Let $w$ be a group word with variables $\bar x, \bar y$, where $\bar x=(x_1,\dots ,x_m)$ and $\bar y=(y_1,\dots ,y_n).$ I am interested in the following questions. (1) Is the sentence $(\forall\bar ...
4
votes
2answers
224 views

Quotients of finitely generated nilpotent groups

Is the following fact true? Let $N$ be a finitely generated nilpotent group, and denote its lower central series by $(N_r)_{r\ge 1}$, that is, $N_1=N$ and $N_{k+1}=[N,N_k]$ is the ...
6
votes
1answer
160 views

Free actions of non-amenable groups

Let $G$ be a locally compact, second countable, non-amenable group, let $X$ be a Haudorff space that is not necessarily compact, and let $G \curvearrowright X$ be a topological action that is free ...
1
vote
0answers
48 views

Dense subgroups in subgroups of profinite groups

Let $G$ be a finitely generated residually finite group and $\hat G$ its profinite completion. Then for all $g\in \hat G$ we have $gGg^{-1}\leq \hat G$ is dense. Suppose that $H\leq \hat G$ is a ...
6
votes
0answers
130 views

The possibility of a symmetric difference in a torsion-free group

Is there a torsion-free group containing two elements $x$ and $y$ and a finite non-empty subset $B$ such that $B=xB \triangle yB$, where $\triangle$ denotes the symmetric difference of two sets and ...
0
votes
1answer
157 views

Find all possible rational pairs of a parametric sextic and all cases where it is reducible for either parameter

Find all possible rational solutions pairs $(z,a)$ of the equation $a^6 + a^4 (-18368 + 9184 z - 2912 z^2) + a^2 (61702144 - 61702144 z + 36814848 z^2 - 10694656 z^3 + 1748992 z^4) - z^2 ...
3
votes
6answers
412 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 ...
4
votes
0answers
149 views

finite approximation equation on free group

An equation in a free group $F$ is an identity of the form $W(x_1,..,x_n,a_1,...a_n)=1$ where $x_1,...,x_n$ are variables, $a_1,...,a_n$ free generators of $F$ and $W$ a word in the free group on ...
9
votes
1answer
244 views

Nontrivial finite group with trivial cohomology in prescribed degree

For any non-trivial finite group $G$ there exists some $j > 0$ such that $H^{aj}(G) \neq 0$ for all $a = 1,2,3,\dots$, see e.g. this question: Non-vanishing of group cohomology in sufficiently high ...
4
votes
0answers
80 views

Meager subgroups of compact groups

Suppose we have an infinite compact (Hausdorff) group $G$, and a subgroup $H\leq G$ which is meagre. Can $H$ always be covered by a countable family of nowhere dense sets $H_n$ such that $H_n^2$ is ...
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 = ...
38
votes
4answers
1k views

On sentences true in all finite groups

Let $w$ be a group word with two variables $x$ and $y$. Is the sentence $(\forall x)(\exists y)w=1$ true in every group if it is true in every finite group? The same question about the sentence ...
-1
votes
1answer
209 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
82 views

A subgroup of corank 1 in a free group contains a primitive element?

Let $F$ be the free group on $\{x_i\}_{i=1}^\infty$, and let $H \leq F$ be a subgroup with $\langle H \cup \{x_1\} \rangle = F$. Must there be a free basis $B$ of $F$ for which $B \cap H \neq ...
4
votes
0answers
163 views

Primitive elements in a free group

Let $F$ be a free group of rank $\aleph_0$ and let $H$ be a subgroup for which there exists some $a \in F$ such that $\langle H \cup \{a\} \rangle = F$. Must there be some free basis $B$ for $F$ and a ...
3
votes
1answer
134 views

Preprint by Wall on Sjogren's theorem

In their account http://dx.doi.org/10.1016/0022-4049(87)90048-X of Sjogren's theorem, Cliff and Hartley refer to two articles: [9] B. Hartley, A note on a lemma of Sjogren relating to. dimension ...
0
votes
1answer
147 views

Must a group of defficiency > 1 be nonabelian?

Let $F$ be a free group of finite rank $n > 2$, and let $S \subseteq F$ be a subset of cardinality at most $n-2$. Denote by $S^F$ the normal subgroup of $F$ generated by $S$. Must $F/S^F$ be ...
4
votes
1answer
152 views

Nilpotent of class 2 free product

Question. How is the nilpotent of class 2 (nil-2) free product of groups defined? I came across this construction reading the following paper. Alan H. Mekler (1981), Stability of nilpotent groups of ...
5
votes
0answers
105 views

Example of a torsionfree group satisfying a cohomological condition

Let us call a finitely generated group $G$ cohomologically rich if for each $k \geq 0$, we can find a subgroup $G'$ and a prime $p$ such that $H^k(G';\mathbb F_p) \neq 0$. Examples which come to mind ...
4
votes
3answers
201 views

Enumerating cosets of the modular group

Suppose we chose the generators $f(z) = z+1, g(z) = z-1, h(z) = -1/z$ for the modular group $\Gamma$ (i.e. the group of fractional linear tranformations $z \mapsto (az +b)/(cz+d)$ with $a,b,c,d \in ...
-1
votes
1answer
156 views

Consequences of Serre's property FA

Proposition 21 of Serre's Trees: Let G be a group with property FA. If G is contained in an amalgam then G is contained in a conjugate of one of the amalgam's factors. Can anybody help with this ...
3
votes
0answers
82 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, ...
9
votes
0answers
256 views

Proof of Cauchy's Theorem from Group Theory - Generalizable?

There are many proofs for Cauchy's Theorem from group theory, which states that if a prime $p$ divides the order of a finite group $G$, then $\exists g\in G$ of order $p$. Recently I've encountered ...
2
votes
0answers
91 views

Generating free groups by small subgroups and an element

Let $F$ be a free group of countably infinite rank, and let $L \leq F$ be a finite index subgroup. For some prime $p$ and $k \in \mathbb{N}$, let $\sigma \colon L \to ...
4
votes
1answer
123 views

K-fellow traveler property and automatic structure

I have been reading several articles about automatic groups and metric spaces of negative curvature. However it is not clear for me the relationship between automatic groups, hyperbolcity and the ...
5
votes
1answer
318 views

Elementary equivalence of the direct product and direct sum of groups

It is well-known that the direct product of any family of abelian groups is an elementary extension of the direct sum of the family (see e.g. Lemma A.1.6 in the book `Model Theory' by W. Hodges, ...
6
votes
1answer
90 views

Sum identities with immanants

For $\chi$ being an irreducible character of the symmetric group $S_n$ and being $M$ a complex $n\times n$-matrix, I would like to show $$ \sum_{\sigma, \rho \in S_n} \overline{\chi(\sigma)} ...
1
vote
1answer
82 views

Are Carter subgroups nilpotent projectors?

R. Carter prooved that in finite soluble groups $G$ Carter subgroups $C$ exist and that they are conjugated. Furthermore they are exactly the nilpotent projectors: For every normal subgroup $N$ of ...
14
votes
0answers
237 views

Group with finite outer automorphism group and large center

Does there exist a finitely generated group $G$ with outer automorphism group $\mathrm{Out}(G)$ finite, whose center contains infinitely many elements of order $p$ for some prime $p$? A motivation is ...
2
votes
0answers
69 views

Splitting over infinite generated abelian subgroup?

Recall that a group is CSA if all its maximal abelian subgroups are malnormal. Question 1: Can a finitely generated CSA group splits (as graph of groups) over an infinitely generated abelian ...
4
votes
2answers
282 views

Relation between Associative algebra and group algebra

Let $A$ be an associative algebra over a filed $k$. Q) What are the condition we can impose on $A$ such that there exists a $G$ such that $A=k[G]$, the group algebra generated by $G$? I am ...