Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
8,182 questions
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Maximal subgroups of PSL(n,q)
Dear All,
Does someone know some source for detail description of maximal subgroups of $PSL(n,q)$ , and their indices.
Thanks in advance.
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A problem with Neumann's lemma.
In a paper I am reading at the moment (Hrushovski Martin, elimination of imaginaries in $Q_p$), in some proof they use the following fact (at least this would be enough to get their proof going, but ...
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1-Parameter subgroups
Let $G$ be an affine reductive group defined over a field of characteristic zero $k$, denote by $\bar{k}$ the algebraic closure of $k$, and by $k^{\times}$ the multiplicative group of $k$. Let $Z(G)$ ...
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Reductive groups question
Considering a complex algebraic group G defined over the reals, one knows from an article of Borel and Harish-Chandra (Arithmetic subgroups of algebraic groups, Annals of Mathematics 75 (1962)) that G ...
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Name for number of elements of order (1 or) 2 in an abelian group
Let $G$ be an (additive) abelian group. Is there a concise, standard term for the order of the kernel $K$ of the endomorphism $g\mapsto 2g$? Or alternatively, for the index of $K$?
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Subgroups of the union of conjugates
This question is an attempt to find a version of this question with a positive answer. In this question, we ask if one can find big subgroups in the union of conjugtes of small subgroups of finite $p$-...
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Under what conditions there is a one-to-one mapping between a product of matrices and the sequence of matrices leading to the product? [closed]
I have a set of matrices $A_1,\ldots,A_n$. Let $\mathcal{A} = \{A_i\}$.
What are some simple conditions under which for any sequence of indices between $1$ and $n$, $a_1,\ldots,a_m$, the product
$$\...
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Profinite completion
What is the profinite completion of the group $S^1$?
where $S^1= \{ z\in\mathbb{C}: |z|=1 \}$ is a compact and abelian group.
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How does Sage order the elements of the symmetric group?
In Sage, the symmetric group is a list. For instance if G = SymmetricGroup(3), we have
\begin{align*}
G[0] & = e \\
G[1] & = (1,3,2)\\
G[2] & = (1,2,3) \\
G[3] &= (2,3)\\
G[4] &= (...
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Must finite groups with isomorphic commutators and quotients be isomorphic?
Let G and H be finite groups. Let G' = [G,G] and H' = [H,H] be the corresponding derived groups (commutator subgroups) of G and H. I am looking for an example where G' is isomorphic to H' and G/G' is ...
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Splitting of a finite group with no abelian subfactor in composition series
Let $G$ be a finite group with no abelian subfactor in its composition series.
Is $G$ obtained from simple groups by iterating semidirect products?
(Initially it was asked whether $G$ is a direct ...
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The X-series (for groups)
It goes without saying that the name in the title tentatively refers to a series whose name one does not know yet and probably in the future I may come with a post titled "The x-sequence" or "The x-...
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Is there a permutation group satisfying the following property?
Let $n=3^m$ for some positive integer $m$. Let $G\leq S_n$ be a transitive permutation group on $n$ letters. Denote the largest normal subgroup of $G$ with odd order by $O_{2'}(G)$. My question is the ...
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about simple non-abelian 2-generated group [closed]
Does there exist a simple non-abelian 2-generated group $G$ and two elements $a, b \in G$, such that $\langle \{a, b\} \rangle = G$, $a^2 =1$ and $\forall c, d \in G$ $\langle \{c^{-1}bc, d^{-1}bd \} \...
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coloring infinite vertex transitive graph without large cliques
Let $G$ be an infinite vertex-transitive graph (this means that for every $u,w \in V(G)$ there exists an automorphism $\tau$ of $G$ such that $\tau(u) = v$).
We assume that $G$ is undirected, and does ...
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Direct limit of lattice-ordered groups
In general, any abelian group can be expressed as a direct limit of its f.g. subgroups. For the case of $\ell$-group (lattice-ordered group) is that true or not? As an abelian group we do not have ...
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Sets with compatible right and left G-actions
Given a group G acting both on the left and the right of a set X, say the actions are compatible if $g \cdot (x \cdot g') = (g\cdot x)\cdot g'$ for all $g,g'\in G$ and $x\in X$.
Here are some ...
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Center of a Symmetric Group on an Infinite Set
Let $X$ be a set, $G$ the group of bijection on $X$. Then it is well-known that if $|X|\geq 3$, $Z(G)$ is trivial. However, I cannot see a way of extending this proof to $X$ being an infinite set (...
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Trans-universality for finitely generated groups
QUESTION: does there exist a group U such that three conditions hold:
(a) every finitely generated group is isomorphic to a subgroup of U;
(b) for every group G that is not finitely generated there ...
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A question on permutation groups
Let $a_1$, $a_2$, and $a_3$ be three involutions of a finite set such that $a_1 a_2 a_3$ is a cyclic permutation. Is the group generated by $a_1, a_2, a_3$ the symmetric group?
user513682
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Are there overwhelmingly more finite posets than finite groups? [closed]
A function $f:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ overwhelms $g:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ if for any $k\in \mathbb{Z}_{\geq 1}$ the inequality $f(n)\leq g(n+k)$ holds only for ...
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Can finite index be seen at the level of profinite completion
Let $G$ be a group, and $H$ a subgroup of $G$.
Is it possible to "see" from the profinite completions of $H$ and $G$ that $H$ has finite index in $G$?
Naively, does $H$ have finite index in $G$ iff ...
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How can we pave the multiplicative semigroup $(\mathbb N,\cdot)$?
Let $(S,\cdot)$ be a semigroup and $W\subseteq S$ be a subset. Let me call $W$ "tile" if the following property is satisfied: there exist $s_1,...s_k\in S$ such that the sets $s_i\cdot W$ are pairwise ...
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Is the cross-product of two subgroups another subgroup (as claimed in the following paper)?
Hi..
In the second paragraph of the following paper, there is a statement: "Because the direct product of subgroups is automatically a subgroup.."
http://jmp.aip.org/jmapaq/v23/i10/p1747_s1?...
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Can we generalise groupoids to monoid-oids? [closed]
Groups correspond to one object categories where every morphism is an isomorphism. Monoids correspond to one object categories.
Groupoids correspond to small categories where every morphism is an ...
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Faithful irreducible representations of the dihedral group $D_{2p}$, of dimension at most $p-1$
I am curious about the irreducible representations $\rho: D_{2p} \rightarrow GL_n(\mathbb{Q})$ of dimension at most $p-1$, not the real or complex representations. My mind is occupied with these two ...
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Dimension of irreducible representation associated to a Young tableau
This might be a classic question, but since I am new to representation theory of the symmetric group, I am asking it here.
Suppose that $\lambda_1 \geq \lambda_2 \geq \dots \lambda_k$ and $\rho$ be ...
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Group actions on affine space which are almost good
Let $G$ be a finite group acting on $\mathbb A^n_{\mathbb C}$. Let $Y$ be a dense open whose complement is of codimension at least two.
Assume $Y$ is $G$-stable, the action of $G$ is free on $Y$, ...
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Which finite cyclic groups can be characterized by Lattice isomorphism and isomorphism between their automorphism groups?
Given a finite cyclic group $G$, we denote by $L(G)$ the lattice of its subgroups, and by $\mathop{\rm Aut}(G)$ the automorphism group of $G$. Let $H$ be any group. Assume that $L(G)\cong L(H)$ and $...
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Does $\mathbb Z \times \mathbb Z$ mod the obvious $\mathbb Z$ action have more structure than just a set?
$\mathbb Z$ acts on the lattice $\mathbb Z \times \mathbb Z$ by adding an element to itself n times.
I am studying some function arising from symplectic geometry which happens in my case to be ...
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For which semisimple element $s$ in finite group of lie type centralizer $C_{G}(s)$ of $s$ is a Levi subgroup ?
Let $G$ be a finite simple group of lie type. Let $s$ be a semisimple element lying in maximal torus $T_{w}$ for $w\in W$ where $W$ is the Weyl group of $G$. Can we say that $C_{G}(s)$ is Levi ...
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decreasing chain of subgroups in the Heisenberg group
By Heisenberg group I mean the group with presentation $H$ generated by $x$ and $y$ such that $x$ and $y$ commute with $xyx^{-1}y^{-1}$. Is there an infinite chain of subgroups
$H > H_1 > H_2 &...
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Finitely generated and finitely presented [closed]
For a group, it seems fairly clear that finitely presented implies finitely generated.
But what about the converse? Is there a finitely generated group that is not finitely presented.
(Let's say ...
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Generators of $SL(n,\mathbb F_2)$? [closed]
Consider the invertible matrices in $\mathbb F_2^{n\times n}$ which are a multiplicative group structure. Is there a finite set of $2k$ (at a $k\in\mathbb Z_{\geq1}$ independent of $n$) generators for ...
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the number of minimal generating subsets of a group
Clearly every finite group has a minimal generating subset.
Is there any formula for the number of minimal generating subsets of a finite group?
Is it known which groups have a unique minimal ...
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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 ...
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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: (...
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Unipotent orbit in adjoint group over finite field
[Editted: The assertion is wrong; see Jay's answer]
My apology if this question is too simple. I am reading Deligne-Lusztig "Reductive groups over finite fields" and at the beginning of Chap. 4, ...
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The number in the join of conjugate class and centralizer
I want to know that whether follow equality holds:
$ |N_G(C_G(a)):C_G(a)|=|a^G\cap C_G(a)|.$
It is easy to see that the left hand is no more than the right hand. I think this equality does not ...
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Maximal subgroups of abelian groups and Q-algebras
Let $G$ be an abelian group which does not have a maximal subgroup. Does it follow that $G$ is a $\mathbb{Q}$-algebra?
It is easy to see that $\mathbb{Q}$-algebras do not admit any maximal subgroups. ...
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Sp(2n) intersect Sp(2n,H)? (Please read for explanation of notation)
First let me fix some notation:
Let $O(n)$ be the group of $n \times n$ real matrices $T$ which are "orthogonal", $U(n)$ be the group of $n \times n$ complex matrices $T$ which are "unitary" and $Sp(...
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Torsion-free subgroup of affine group
Let $G$ be a finitely generated group and $\varphi:G\to \operatorname{Aut}(\mathbb C)$ a homomorphism, where $\operatorname{Aut}(\mathbb C)$ is the group of complex affine transfromations $a z+b$.
Can ...
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Are all Coxeter groups virtually free or virtually surface groups?
From Surface subgroups of Coxeter and Artin groups (Gordon, Long and Reid, 2003) DOI link, we can read that (Theorem 1.1) a Coxeter group is either virtually free or contains a surface group ($\pi_1$ ...
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Example of a representation of a finite group where Weyl's unitary trick is necessary?
Is there an example of a representation $\rho: G \rightarrow GL(V)$ for some finite group $G$ where say $W \subset V$ is a $G$-invariant subspace for $\rho$ but the orthogonal complement (in the ...
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$\text{Max}\big(\text{Sub}(\text{Sym}(\omega))\setminus \{\text{Sym}(\omega)\}\big)$
If $G$ is any group, then by $\text{Sub}(G)$ we denote the collection of all subgroups, ordered by $\subseteq$. If $(P,\leq)$ is a partially ordered set we let $\text{Max}(P)$ and the set of maximal ...
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simple groups all sylow subgroup is nonabelian
Thanks for any help or comments
How can I find the list of all non abelian simple groups (particularly simple lie type) such that all $p$-Sylow subgroups are non abelian for odd prime $p$?
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When are groups subgroups of a same group?
I put this question on Stackexchange :
https://math.stackexchange.com/questions/1659760/when-are-groups-subgroups-of-a-same-group
but it got no answer, so I post it here.
Let $\mathcal{G}$ be a ...
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4th Order Floretions: Floret's Equation [closed]
Update: I've marked this question as answered. If you are thinking "What the heck are floretions?", go right to the answer provided by the Grinch. I definitely should have added clearer information on ...
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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 ...
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Is there a classification of possible linear actions?
In a vector space, linear transforms can act on points of the space by the usual matrix multiplication rule, but in this note I am reading they use a different action (The Möbius transformation). It's ...
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