# Tagged Questions

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### Question about May's Algebraic Topology book

I am referring Google Books for the question: link in the proof of the first lemma, why is $hns=\phi(hs)$ true? I simply cannot get it...
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### Notation for pointwise versus “setwise” stabilizers

Suppose one is working with both pointwise and setwise stabilizers of sets under a group action. Are there common conventions for notationally distinguishing these two notions? How common are they? ...
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### Is the stabilizer of an element $\delta$ in the stabilizer of $\omega$ in G equal to the pointwise stabilizer of $\{ \delta, \omega \}$

i.e., is $(G_{\delta})_{\omega} = G_{( \{\delta, \omega\} )}$? I know that \begin{eqnarray*} (G_{\delta})_{\omega} &=& \{ \forall g \in G_{\delta} \,|\, \omega^g = \omega \} \\ &=& ...
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### How to prove that $N$ is 2-transitive on $\Omega$?

Suppose $\Omega$ is a finite set with $|\Omega| \geq 5$. Let $G$ act faithfully on $\Omega$ such that $G$ is 4-transitive on $\Omega$. Let $N$ be a normal, nontrivial, nonregular subgroup of $G$. I ...
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### a question concerning subgroup of symmetric group

Suppose $H$ is a transitive subgroup of the symmetric group of $n$ symbols. Show that $n$ divides the order of $H$. I tried to show that some $n$-cycle is in $H$ but this idea did not work.
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### Show that group action is homomorphism to Symmetric group

I'm just barely getting my feet wet with abstract algebra, currently working on understanding group action. According to the wikipedia article, a group action $A$ of group $G$ on set $X$ is a group ...
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### the cardinal of $x^G$ factors the cardinal of $x^N$

please give me hints to solve this problem: Let $G$ acts on $X$ and $N$ be a normal subgroup of $G$, show that for every $x\in X$ we have: the cardinality of $x^G$ factors the cardinality of $x^N$ .
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### Transitive group action restricted to normal subgroup

Let $G$ be a ﬁnite group, and let $\Omega$ be a transitive $G$-space. Assume 1 $\neq H \unlhd G$ and that |$\Omega$| = $p$ where $p$ is prime, and $G \leq Sym(\Omega)$. Deduce that then $H$ must act ...
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I proved the following proposition as an exercise: Suppose $H \leq K \leq G$ are groups and that $G$ acts on $\frac{G}{H}$ and $\frac{G}{K}$. If $H$ is subconjugate to $K$ (i.e., if $\exists g \in ... 2answers 89 views ### Group Action Questions? We discussed Group Actions in my undergraduate Modern Algebra class today. I understand the definition and example we went over in lecture, but the problem set is proving difficult. If I want to ... 1answer 43 views ### Group action with finite stabilizer. Let$G$be a group generated by$\{g_1,g_2,\ldots , g_n\}$. Let$X$be a space with a$G$-action on it, i.e.$G$is acting on$X$. Suppose for each$x\in X$, the set$\{g_i;g_i(x)=x\}$is trivial. ... 1answer 97 views ### Two subgroups$H_1, H_2$of a group$G$are conjugate iff$G/H_1$and$G/H_2$are isomorphic Let$H_1$and$H_2$be subgroups of some group$G$. Prove that the left$G$-sets$G/H_1$and$G/H_2$are isomorphic (as left$G$-sets) iff the subgroups$H_1$and$H_2$are conjugate. If$H_1$... 1answer 57 views ### Action on Pairs, On Sets and on points in GAP I am trying to understand GAP in group action. I am confused in few things what is the difference between action on pairs, on sets, with the domain sometimes on list, and on blocks. Please help me to ... 2answers 193 views ### About stabilizer in group action Let$X$be a finite set and$x$is an element of$X$. Let$G_x$, the stabilizer subgroup, be the subset of$S_X$consisting of permutations that fix$x$. The question is Is stabilizer always a ... 1answer 67 views ### When is the action of$G$on$\text{Syl}_p(G)$by conjugation is double transitive? We know that the action of$G$on$\text{Syl}_p(G)$by conjugation is transitive. I wonder when this action can be double transitive on$\text{Syl}_p(G)$. Thanks for your help. 3answers 237 views ### Without using Sylow: Group of order 28 has a normal subgroup of order 7 Prove that a group of order 28 has a normal subgroup of order 7. How can I prove this without using Sylow's theorem? I know by Cauchy’s theorem, there exists an$x\in G$with order 7, now I just ... 3answers 65 views ### Relationship between group actions and homomorphisms I know that there exist no nontrivial homomorphism from$S_3$into$Z_5$as they are groups of co-prime order. I am not looking for an explanation of this but for an explanation concerning the obvious ... 0answers 30 views ### invariance of 2-form under$SO(3)$I'm trying to understand how to derive forms that invariant under action of some group. For example 2-form on$S^2$and on$\mathbb{R}^3$is very interesting for me (because I have troubles with it). ... 0answers 57 views ### Terminology on group actions Johnson, D. L. "Minimal permutation representations of finite groups." Amer. J. Math. 93 (1971), 857-866. My knowledge of group theory is undergraduate-level stuff. I'm looking at the paper cited ... 1answer 92 views ### Understanding what an action is? This is a very simple question, and I am quite embarrassed to ask it! I'm trying to understand what an action is in general, and perhaps the best place to start is to try and outline my current ... 1answer 88 views ### Orbits of$\mathbb{Z}_n^{*}$acting on a set$\mathbb{Z}_n$Let$n\geq 2$be an integer and consider the action$\Phi: \mathbb{Z}_n^{*}\times \mathbb{Z}_n \rightarrow \mathbb{Z}_n$defined as $$\Phi(\alpha)(x)=(\alpha x \textrm{ mod } n),$$ i. e. simply the ... 1answer 29 views ### Defining a map based on a group action on left cosets If$H$is subgroup of$G$such that the index of$H$in$G$is$n$and$\pi_H$is the permutation representation of the action of$G$on the left cosets of$H$, is$\pi_H$a map from$H$to$S_n$? I ... 1answer 66 views ### Orbit and Stabilizer Are the following definitions essentially the same: Orbit: Let$G$be a group of permutations of a set$S$. For each$s \in S$, let$\operatorname{orb}_G(s)= \{f(s) \mid f \in G\}$. The set ... 1answer 52 views ### Question about primitive group actions In Glass' Partially Ordered Groups Corollary 7.4.4 says: If$G$is an ordered group and$(G,G)$is the right regular representation, then$(G,G)$is primitive if and only if$G$is ... 2answers 86 views ### Using counting formula to get |G| = |kernel φ||image φ| The counting formula I am saying : Let S be a finite set on which a group G operates, and let Gs and Os be the stabilizer and orbit of an element s of S. Then |G|=|Gs||Os| or ... 1answer 106 views ### Classification of transitive G-sets for a given group of small order Given a group of small order (<30), how does one go about systematically finding all the transitive G-sets up to isomorphism? By X and Y being isomorphic we mean there are maps$f:X \rightarrow Y$... 0answers 79 views ### Stabilizer map on transitive G-set defines a morphism with G acting on subgroups by conjugation This is part of a homework problem for a graduate course on abstract algebra. Given a transitive G-set$X$, show that the map that assigns to$x \in X$its stabilizer defines a morphism of G-sets ... 1answer 64 views ### Is the kernel of this group action the centralizer? In Dummit and Foote, they state "... let the group$N_G(A)$(normalizer) act on the set$A$by conjugation. It is easy to check that the kernel of this action is the centralizer$C_G(A)$." From ... 0answers 121 views ### Why are they called orbits? When we study actions in group theory, we consider sets of the form $$\text{Orb}_G(x)=\{gx\mid g\in G\}$$ that are called orbits. Although, the only reason I find convincing for that name is that in ... 1answer 23 views ### Modules over a group presented via a free group. Say$G$is presented via a free group$F$freely generated by$S=\{s_i, 1=1,2,\dots\}$. Then$\pi:F \rightarrow G$the canonical projection. Let$R$be any commutative ring. Can we follow that any ... 1answer 102 views ### Find Orbit of element$m \in M, M = M(2, \mathbb{R})$under action of group$G = GL(2, \mathbb{R})$mapping$m$to$g^{-1}mg$,$g \in G$. Find Orbit of element$m \in M, M = M(2, \mathbb{R})$under action of group$G = GL(2, \mathbb{R})$mapping$m$to$g^{-1}mg$,$g \in G$. The element$m$is$ \left( \begin{matrix} 2 & 1 \\ 0 ...
Let X be a set of order n. a) If G acts transitively on X then n divides $| G |$. b) If G acts 2-transitively on X then n(n-1) divides $| G |$ For a) i first prove that if G acts transitively on ...