Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
8,182 questions
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What do we know about this ideal of the group algebra?
Let $G$ be a torsion-free amenable group. Consider, $\mathbf M$, the collection of all multiplicative functionals on $\mathbb CG$, the complex group algebra of $G$. So, $\ker\phi$ is an ideal of $\...
- 2,118
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Greatest common divisor of two specified sequences of numbers (search for equality)
I consider two sequences of numbers $A=\{a_1,...,a_n\}$ and $B=\{k-a_1,...,k-a_n\}$, where $a_1 \le a_2 \le ... \le a_n \le k$.
I am looking for such conditions under which: $\gcd(a_1,...,a_n) = \gcd(...
- 13
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A question concerning some group action
Let $G$ be a finite group. Consider the set
$$X = \bigcup_{H \le G} G/H$$
which is a disjoint union of left cosets of subgroups $H$ of $G$.
Then $G$ acts on $X$ by left multiplication, and the number $...
user6671
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Absolute center of finite $p$-group
For any group $G$, the absolute center $L(G)$ of $G$ is defined as
$$L(G) = \lbrace g\in G\mid \alpha(g)=g,\text{ for all }\alpha\in Aut(G)
\rbrace,$$ where $Aut(G)$ denote the group of all ...
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72
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Isomorphism of finite groups and cycle graphs
Let $G$ and $H$ be finite groups and suppose they do have the same cycle graph. Is it possible to argue that this implies $G$ and $H$ are isomorphic? If yes, why? If not, is there an explicit ...
- 489
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345
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Normal subgroups of $p$-groups
I was reading Professor Yukov Berkovich' paper "On Subgroups of Finite $p$-groups" when I stumled upon the following theorem:
Let $G$ be a nonabelian $p$-group with cyclic Frattini subgroup, $|\Phi(...
- 415
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Groups implementable by finite field
I'm interested in finding all groups for which the group operation (and inverse map) may be implemented using finite field arithmetic.
I've done some searching and have come across "algebraic groups",...
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179
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Monomial Characters of Quotient Groups
The following statement provides (if true) a powerful tool for inductive proofs. Can anyone confirm if it is true:
Suppose $G$ is a finite group and $N$ a normal subgroup of $G$. Is it true that if $...
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$2$-power-torsion elements of a group
Let $G$ be a finite group, let $P$ be one of its $2$-sylow subgroups. Let $H$ be a proper subgroup of $P$, namely $H<P$ with $H\neq P$. Is it possible that $$\bigcup_{g\in G}g^{-1}Hg=\bigcup_{g\in ...
user90517
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Coxeter group action on the product of root systems
Let W be a finite Coxeter group and $\Phi^+$ the set of its positive roots. The Coxeter group acts on $\Phi^+$ by $(w, \alpha) \mapsto w \cdot \alpha$ if $w \cdot \alpha \in \Phi^+$ and $(w, \alpha) \...
- 6,201
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Profinite groups with finite torsion
Let $G$ be a profinite abelian group such that for every $x\in G$ and every $n\in\mathbb Z$ the preimage of $x$ under the multiplication by $n$ map is finite. Does it follow that the torsion subgroup ...
- 2,305
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Groups with cyclic radicals
Let $G$ be a torsion-free group. For an element $g \in G \setminus \{1_G\}$ we define the radical of $g$ in $G$ as
$$
Rad_G(g) = \left\{r \in G \mid r^a = g^b \mbox{ for some } a,b \in \mathbb{Z}\...
- 191
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A particular functor on the category of abelian groups?
Is there a functor $F$ from the category of abelian groups to itself such that for every non trivial group $G$, $F(G)$ can not be embedded in $G$?
Edit: According to the comment by Prof. Goodwillie ...
- 356
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140
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Finite group and cyclic cover
Suppose the finite group $N$ surjects to finite group
$F$. It is true that for any $G = N ⋊_α \mathbb{Z}$ there are infinitely many covers of $G$ that are cyclic and
surject to $F$.
But is this ...
- 1,755
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328
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Find the trace for some elements in group algebra
Let $K=\langle b,c,d\mid b^{2}=c^{2}=d^{2}=bcd=1\rangle $. Now we consider $$D=K*\mathbb Z/2\mathbb Z=\left\{a,b,c,d\mid a^{2}=b^{2}=c^{2}=d^{2}=bcd=1\right\}$$ where $*$ is the free product. Then we ...
- 407
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Intersection of two subspaces of a Hilbert space
Background:
Let $D$ be a Klein Four group and consider free product $Z/2Z\star D=<a,b,c,d|a^{2}=b^{2}=c^{2}=d^{2}=bcd=1>$. Now we consider group algebra generated by $Z/2Z\star D$ with inner ...
- 407
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Are there some references about a result of inversion set?
Let $w \in S_n$ and $inv(w) = \{(i,j): i,j \in \{1,\ldots,n\}, i<j, w(i)>w(j)\}$ the inversion set of $w$. Let ${\bf i}=(i_1,\ldots,i_m)$ be a sequence such that $s_{i_1}\cdots s_{i_m}$ is a ...
- 6,201
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Characterisation of normality in direct products
Let $G = A \times B$. Suppose that $H \leq G$ such that $N_G(H) = N_A(\pi_A(H)) \times N_B(\pi_B(H))$ where $\pi_A$ and $\pi_B$ are the respective projection homomorphisms
For simplicity and ...
- 366
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Pronorm of a finite solvable group
Let $G$ be a group. The pronormaliser of a subgroup $H$ in a group $G$, denoted $P_G(H)$, is defined to be the set of elements of $G$ that pronormalise $H$. That is, $$P_G(H) = \{g \in G \; | \; \...
- 366
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Case in which the pronormaliser of a subgroup is a subgroup of the group
The pronormaliser of a subgroup $H$ in a group $G$, denoted $P_G(H)$, is defined to be the set of elements of $G$ that pronormalise $H$. That is, $$P_G(H) = \{g \in G \; | \; \exists x\in \langle H, ...
- 366
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Pronormaliser of a subgroup
The pronormaliser of a subgroup $H$ in a group $G$, denoted $P_G(H)$, is defined to be the set of elements of $G$ that pronormalise $H$. That is, $$P_G(H) = \{g \in G \; | \; \exists x\in \langle H, ...
- 366
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Inner products on abelian groups and general modules
Where can I find a discussion about inner products on something more general than vector spaces, and to which extent should one attempt such a generalization?
My particular interest is in abelian ...
- 229
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A question on minimal subgroups
I don't know whether it is a good title for this question or not. Please if possible suggest a different title.
Let $G$ be a finite group and $L$ be a minimal subgroup of $G$ of order $p$ for some ...
- 479
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Character theory of finite groups
Let $G$ be a finite group, $R(G)$ be the solvable radical of $G$, and $G/R(G)$ be isomorphic to $PSL_2(11)$. Also let $\lambda$ be a non-trivial complex linear character of $R(G)$ such that $\lambda$ ...
- 841
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Soluble groups with Min-n
Are there any uncountable soluble groups satisfying Min-n (the minimal condition on normal subgroups) other than the group constructed by B. Hartley in [Uncountable Artinian modules and uncountable ...
- 379
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Non-normal subgroups of triangle groups
Consider the triangle group $\Delta(a, b, c)$ for some $a, b, c > 1$, given by three generators $x$, $y$, and $z$, such that $x^2 = y^2 = z^2 = (xy)^a = (yz)^b = (zx)^c = 1$.
Can anyone provide an ...
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Infinite groups in which every element is a commutator
Finite groups in which every element is a commutator are considered in many works. How about infinite group case? Are there any recent results or constructions related to infinite groups in which ...
- 379
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Simple automorphisms of finite relations
Finding automorphisms is a hard problem in general, but I am studying some simple subgroups of automorphisms, which are easy to find.
I have some r-ary relation R on a finite set U (if it was a ...
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specific qi on free groups
Let $F_n$ be the free group on $n$ generators, $n>1$.
If $\phi$ is a quasi-isometry (or a bijective bilipschitz equivalence) on $F_n$, then what can we say about the explicit form of $\phi$?
In ...
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minimal permutation representations [duplicate]
Suppose I have a finite group $G.$ How hard is it to find the (a?) minimal degree permutation representation of $G?$ The second part of the question is: is there a table of such (hopefully for ...
- 96.4k
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Isometry group of a complete separable metric space is Polish?
Let $(X,d)$ be a complete separable metric space, and endow $Iso(X,d)$ with the pointwise convergence topology.
I've seen a few sources say this is clearly a Polish group, but why is this this the ...
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If $G$ is profinite and $M$ finite abelian, must there exist an open subgroup $H\le G$ with $H^2(H,M) = 0$?
Let $G$ be a profinite group, and $M$ a finite abelian group with trivial $G$-action. Must there exist an open subgroup $H\le G$ with $H^2(H,M) = 0$?
- 7,527
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Embedding a cancellative monoid into another in such a way that $|X-x|=|X|$, where $X$ is a fixed finite set and $x\in X$
Preliminaries.
Let $\mathbb A = (A, +)$ be a possibly non-commutative semigroup. For $X, Y \subseteq A$ we set
$$
X - Y := \{a \in A: a + y \in X\text{ for some }y \in Y\},
$$
which is just the usual ...
- 10.5k
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generalized word problem
If $H$ is a finitely generated subgroup of $G$ and if $H$ given by say a finite set
of words which generate it, then the generalized word problem for $H$ in $G$
is the problem of deciding for an ...
- 421
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145
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Group which is not MF or AF
Does someone know example of group (countable, discrete) which can not be embedded (monomorphism) into
$$ U(\prod M_n/\oplus M_n)$$
unitary group of universal MF-algebra? Or example of group which can ...
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Basic question about power series and complete group algebras
This is a pretty basic question, but I suspect it might be too exotic for math.stackexchange.
Let $\mathbb{Z}_p$ be the $p$-adic integers. For free pro-$p$ group $F_r$ of rank $r$, we can consider ...
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Computational Algebra and Symbolic Computation - Where? [closed]
Following the line of this question, I'm in my last year of M.Sc., and I'm looking for a place where I can start my PHD. Since that question has been asked 4 years ago, I thought it may be wise to ask ...
- 201
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Abelian centralizer groups (CA-groups)
I am searching for all information about CA-groups [abelian centralizer groups] and i just found a German book [Huppert] and Nilpotent Centralizer group of Suzuki in 44 pages and Group theory book of ...
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When is a group generated by three involutions two of which commute a Coxeter group?
Let's say the group is generated by three involutions $a$, $b$, and $c$ such that $ord(ab)= 2$, $ord(bc)=3$, and $ord(ac)=m$. Under which conditions is it isomorphic to the rank 3 Coxeter group $(2, 3,...
user76098
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Chinese remainder theorem for cyclic subfactor planar algebras
This post was inspired by an exchange with the indian woman mathematician Ajit Iqbal Singh.
The chinese remainder theorem can be stated as follows:
Let $n_1, \dots, n_r \ge 2$ be positive integers ...
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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 ...
- 211
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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 ...
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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 \...
- 11.3k
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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 = \...
- 11.3k
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199
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Action of semidirect products of cyclic groups
Is there anything known about group actions of $C_{p}\rtimes C_{p}^{*}$ on the ring of real polynomials $\mathbb{R}[X_{1},\ldots,X_{n}]$, where $C_{p}$ denotes the cyclic group of order $p$ and $p$ is ...
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711
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$\text{Hom}(G,\mathbb{Z})$ [duplicate]
Fix a cardinal $\kappa$ and consider $\mathbb{Z}^\kappa$ with componentwise addition and the subgroup $$F_\kappa :=\{g:\kappa \to \mathbb{Z}: \{\alpha\in \kappa: g(\alpha)\neq 0\} \text{ is finite}\}.$...
- 48.9k
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Asymmetry of functions defined on the $n$-th roots of the unity
Let $\mathcal{A} = \{V : \mathbb{U}_n \rightarrow \mathbb{C}\}$ where $\mathbb{U}_n$ is the group of the complex $n$-th roots of the unity. This group naturally acts on $\mathcal{A}$: for any $a \in \...
- 370
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When is edge colored circulant isomorphism polynomial?
Don't understand enough group theory, but two papers
appear to give partial results about an open problem.
Edge colored graph isomorphism is isomorphism which
preserves the edge coloring (the ...
- 25.4k
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259
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Zariski dense subgroups and conjugates
Let $H \leq \mathrm{SL}_3(\mathbb{Z})$ be a finitely generated subgroup which is not Zariski dense, and let $g \in \mathrm{SL}_3(\mathbb{Z})$. Must there be some $a \in \mathrm{SL}_3(\mathbb{Z})$ such ...
- 11.3k
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$\kappa$-translatability
I asked the following on MSE a few weeks ago but I did not get any answer :
https://math.stackexchange.com/questions/1039593/carlsons-translatability-are-theses-characterisations-equivalent
Reference ...
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