Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
8,181 questions
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Are isomorphic quotients of abelian groups induced by automorphisms? [closed]
If I have an (abelian) group $G$ and an automorphism $\sigma: G \to G$ then for any subgroup $H$ of $G$ there is an induced isomorphism $G/H \cong G/\sigma(H)$ given by the map $gH \mapsto \sigma(g)\...
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Dixon's Theorem [closed]
I am going through a sketch of the proof of Dixon's Theorem (the probability that two randomly chosen elements of A_n generate A_n -> 1 as n -> infinity) due to M. Liebeck and its underlying idea is ...
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In Galois theory, why solvable groups must have their quotient groups be Abelian? [closed]
The definition of solvable groups can be regarded as two constraints, one is that there must be a sequence of normal subgroups, and the other is that the quotient groups between these sequences are ...
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Notation on wreath product [closed]
While reading some (relatively old) papers on group-theory I encountered the following notations whose meanings I cannot understand:
If $W= G \wr H$ is the (unrestricted) wreath product of $G$ and $H$...
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1
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Number of subgroups of a group of orders $p^3$ [closed]
Let $p$ be a prime number. Is there a formula for the number of subgroups of
$$\mathbb{Z}/p\mathbb{Z}\times \mathbb{Z}/p^2\mathbb{Z}$$
$$\mathbb{Z}/p\mathbb{Z}\times \mathbb{Z}/p\mathbb{Z}\times\...
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On Sylow subgroup of a finite group [closed]
Let $p\mid n$, then by $n_p$ we mean the $p$-part of $n$, i.e. $n_p = p^k$ if $p^k\mid n$ but $p^{k+1}\nmid n$. Let $G$ be a finite group, $M\leq G$ and $P\in Syl_p(G)$. Is It true that $|M\cap P|=|M|...
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Maximal torus and application in prime graph [closed]
I am studying papers about " Prime graph" , for example "Prime graph components of finite groups" [ williams], " Groups with complete prime graph connected components" [ Lucido and moghaddanfar]. ...
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Composition factor of a group which isomorphic to the alternating group of order 7 [closed]
I want to find groups whose composition factor is isomorphic to the alternating group of order 7, which groups have this condiction?
best regards
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How about this book Topological Methods in Group Theory [closed]
Topological Methods in Group Theory witten by Ross Geoghegan
What about this book?
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476
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Countable open subgroup
In a Hausdorff topological group, how can I show that every infinite topological group has a countable open subgroup?
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What do you call continous transformations that preserve the finite group structure?
A number of years ago I studied a preon model (Journal of Mathematical Physics 38:3414-3426, 1997) in which the preons interacted like group elements. I thought it curious that you could sometimes ...
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210
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class structure constants relation
Let $C_{j,k}^l$ ,usually called class structure constants, eg Jansen and Boon and/or JQ Chen, be the number of times the class $l$ is generated from the product of classes $j,k$ and $c_j=c_{-j}$ (a ...
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1
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962
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Maximum element order in $S_n$ [closed]
Denote by $S_n$ the group of permutations of the set $\{1,\ldots,n\}$ with composition as binary operation. Let $m_n$ denote the maximum order that an element of $S_n$ can have. What is the smallest ...
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Which self homeomorphisms preserve measure on a torus, apart from affine? [closed]
Which self homeomorphisms preserve measure on a torus, apart from affines? Affine is the composition of rotation and automorphism. Measure is the Lebesgue measure.
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A generalization of an old group problem [closed]
Here is an old exercise in group theory: (1) If $G$ is a group of order $2n$ with $n$ odd then $G$ is not simple and in fact $G$ has a normal subgroup of order $n$. I am going for one straight ...
user39125
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Factor group isomorphic with Klein four-group [closed]
Let $G$ be a finite solvable group and $N$ be a normal subgroup of it which is an elementary abelian 2-group. Suppose that $G/N\cong \mathbb{Z}_2\times \mathbb{Z}_2$ and $|C_G(x)|=16$ for any $x\in G-...
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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 $G_o^{-1}=\{x^{-1}: x\in G_o\}$)
...
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All group structures on a set with cardinality $\aleph_0$
Assume we consider the additive group $(\mathbb{Z}, 0, +)$. I am wondering what other group structures are there with neutral element 0 fixed? Is there a way to classify them or find them all?
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On the maximal ideal m of the formal power series ring [closed]
Let $A \colon= K[[X_1,X_2,\ldots,X_{\infty}]]$ be the formal power series ring with infinitely many variables over a field $K$. We can represent it also by the following manner$\colon$
\begin{equation*...
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Characterization of some finite cyclic groups [closed]
Given a finite cyclic group $G$ with order $n=p_1^{n_1}p_2^{n_2}\cdots p_r^{n_r}$, where $p_i$s are distinct prime numbers $n_i>1$ for all $i$. Let $H$ be any abelian group. Assume that Aut$(G)≅$...
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101
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Orbit size of an element [closed]
Let $H$ be a normal subgroup of $G$ and assume that $G$ is acting over a set $X$. Let $c$ be some element of $X$, is there any relationship among the size of the orbit of $c$ under the action of $H$ ...
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A common name for a functorial construction of Commutative Algebra?
I am interested whether the following construction naturally appearing in Commutative Algebra has some know and acceped name.
Given a commutative monoid $(M,+)$ and a set $X$, consider the family $F(...
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Simple group of order 504 [closed]
As we know,there are 9 Sylow 2-subgroup in the Simple group of order 504.Can anyone prove it only by Sylow's theorem?
(you can't use knowledge about PSL(2,8))
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A presentation for the group $GL(n,\mathbb{Z}_p)$
Let $n\ge 2$. Let $p$ be a prime and $\mathbb{Z}_p$ denote the finite field with $p$ elements.
I want to know about the presentation for the group $GL(n,\mathbb{Z}_p)$ consisting of its generators and ...
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224
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Do monoid homomorphisms from $X^X$ to a group factor through $\text{Sym}(X)$? [closed]
Let $X$ be a set and let $(X^X,\circ)$ denote the monoid of all maps $f: X\to X$, together with composition. Let $(\text{Sym}(X),\circ)$ be the group of all bijections from $X$ to itself.
Does there ...
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A question on the number of subgroups of symmetric groups
Let $G$ be a finite group. Several recent papers (see e.g. http://www.jstor.org/discover/10.2307/2695441) deal with the following notion: $G$ is called a group with perfect order subsets or briefly, a ...
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Conjugacy classes of $PSL_2(11)$ and $PGL_2(11)$ in $Aut(HN)$
How many conjugacy classes each of $PSL_2(11)$ and $PGL_2(11)$ subgroups are contained in the automorphism group of the Harada-Norton group?
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structure of finite polycyclic groups [closed]
We know that every finite nilpotent group is written as a direct product of its Sylow subgroups.
My question is : can we write finite polycyclic groups as a direct product of some subgroups?
if the ...
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on the solvable groups of order $p^aq^b$ [closed]
We know that if $ p$ is a prime number then $ O^p (G) $ is the smallest normal subgroup of $ G $ such that $ G/O^p (G) $ is a $ p $-group.
Now let $ G $ be a finite group of order $ p^aq^b $ where $ ...
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Is the product of closed subgroups in a locally compact group locally compact? [closed]
Let $A$ and $H$ be closed subgroups of a $\sigma$-compact locally compact group $G$. Assume further that $A$ is abelian. Is the group $AH$ locally compact subgroup in the subspace topology?
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Are there overwhelmingly more finite monoids than finite spaces? [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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