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transitively
transitively, adv. (ˈtrɑːnsɪtɪvlɪ, ˈtræns-, -nz-) [f. prec. + -ly2.] In a transitive manner; in the way of transition. a. Gram. In a transitive sense or construction; with a direct object.1571 Golding Calvin on Ps. vii. 7. 20 The woord might also be taken transityvely for too settle or stablish Davi...
Oxford English Dictionary
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Transitively normal subgroup
Here are some facts about transitively normal subgroups:
Every normal subgroup of a transitively normal subgroup is normal. A transitively normal subgroup of a transitively normal subgroup is transitively normal.
A transitively normal subgroup is normal.
wikipedia.org
en.wikipedia.org
Can I use the verb "vanish" transitively? : r/EnglishLearning
The verb “vanish” is primarily used intransitively, meaning it does not take a direct object. It means something disappears by itself. It's ...
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www.reddit.com
Transitive action A group $G$ acts on a nonempty set $X$. Suppose $H$ is a subgroup of $G$ and $H$ acts transitively on $X$. In a proof, an author claims: If $H$ acts transitively on $X$, then obviously $G$ acts tran...
Second, $gx \in X$ and as $H$ acts transitively on $X$, there is _some_ $h \in H$ such that $gx = hx$.
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If a finite group acts transitively on a set, does its center also acts transitively? > If $G$ is a finite group acts transitively on a set $X$, does the center $Z(G)$ also acts on $X$ transitively? I don't see how ...
Hint: Think about the Symmetric group $S_3$ acting on $\\{1,2,3\\}$. What is the center of $S_3$? In fact, any $S_n$ for $n > 2$ acting on $\\{1,2,\dots, n\\}$ will work. Why?
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If a subgroup acts transitively on a set, then the index of the subgroup equals the index of the stabilizer? I am trying to prove the following: If a subgroup $H < G$ acts transitively on a set $X$, then $[G:H] = [G_x...
For the case where $G, H$ can be infinite: Consider $H< H<G \Rightarrow [H:H_x][G:H] =[G:H_x]$ and $H_x<G_x<G \Rightarrow [G_x:H_x][G: G_x]= [G:H_x]$ . $\Rightarrow [G_x:H_x][G: G_x] = [H:H_x][G:H]$. Since $[G: G_x]=[H: H_x] = |X|$, we're done. Note: Can I assume $[G: G_x]=[H: H_x]$ even if $X$ is n...
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If $G\subseteq S_n$ is a subgroup acting transitively on $\{1,\ldots,n\}$, then a nontrivial normal subgroup $N\subseteq G$ has no fixed points Let $G$ be a subgroup of $S_n$, which acts transitively on $I= \\{1, \ldo...
Let $\,\,\\{1\\}\neq N\triangleleft G\,\,$ , and wlog let us assume $\,\,1\in\\{1,2,...,n\\}\,$ is a fixed point of $\,N\,$. But then for any $\,g\in G\,\,,\,x\in N$ , we get $$g^{-1}xg(1)=1\Longrightarrow xg(1)=g(1)\Longrightarrow g(1)\,\,\text{is a fixed point of } N$$ Well, now use that $\,G\,$ i...
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What non-abelian groups have a minimal permutation representation that acts transitively $\{1,2,\ldots, k\}$? This question asks if a minimal permutation representation $\overline{G}$ of a group $G$ (that is, a subgro...
An example of a nonabelian group that is not a direct product in which the smallest degree faithful permutaion representation is intransitive is the group $\langle x,y \mid x^3=y^4=1, y^{-1}xy=x^{-1} \rangle$ of order $12$. It embeds into $S_7$ with $x \mapsto (1,2,3)$, $y \mapsto (2,3)(4,5,6,7)$. T...
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Prove that $SO(3)$ acts transitively on the unit sphere $S^2$ of $\Bbb R^3$ > Prove that $SO(3)$ acts transitively on the unit sphere $S^2$ of $\Bbb R^3$. I think $S^2$ means a 2-D sphere and $SO(3)$ is the usual $SO...
It suffices to show all of $S^2$ is the orbit of a single point, say $e_1$. The equation $Ae_1=x$ says that the first column of $A$ is $x\in S^2$. To create the rest of the matrix $A$, simply take any two vectors on our sphere that are orthogonal to each other and $x$. To do this, compute $e_1\times...
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G acts faithfully on X, N is a minimal normal subgroup of G, N abelian, acts transitively. Prove G acts primitively I want to ask for a hint or solution to this problem: G acts faithfully on X, N is a minimal normal ...
Show that normal subgroups preserve partitions. Show that in this situation only the trivial partitions can be preserved.
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A cute question on group action Let $G$ be a subgroup of $S_n$ that acts transitively on $(1,2,...,n)$. Let $N$ be a non trivial normal subgroup of $G$. Does $N$ act transitively on the set? Its true when $n$ is prime.
Take any non-simple group $A$, and let it act on itself by left translations. If $A$ has $n$ elements, this allows you to think of $A$ as a subgroup of $S_n$. Now consider any proper non-trivial normal subgroup $B$ of $A$.
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Does group of deck tranformations acts transitively on each fibre if it acts traansitively on one fiber? i am reading bredon "Topology and Geometry " It states that if we have a covering map p : X ->Y s.t. p(x) = y.X...
Let $G$ be the group of deck transformations of the cover $p$. Let $y, y' \in Y$, and let $F, F'$ be the fibres above $y$ and $y'$ respectively. The sets $F$ and $F'$ are $G$-sets, i.e. sets equipped with an action of $G$. I claim that they are isomorphic $G$-sets. Indeed, let $\sigma$ be a path in...
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How big are transitively reduced graphs? What is the largest transitively reduced acyclic connected graph on $n$ vertices, for every $n$? How many edges $e$ does it have? How does $e$ grow as a function of $n$? Faste...
Since the transitively reduced graph $G$ is acyclic, there is a corresponding undirected simple graph $G'$, which can be formed by ignoring the direction For $G$ to be transitively reduced, a necessary condition is that $G'$ must not contain any triangles.
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On what sets can $\mathfrak{S}_n$ act transitively? I would like to know $\mathfrak{S}_n$ could act faithfully transitively on sets with $m$ elements, with $m > n$. I know that it is not possible if $m = n+1$ except f...
There are many sets of more than $n$ elements on which $\mathfrak S_n$ acts transitively. The largest possible example is that of the $n!
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