Question

How to prove that $S_4$ a symmetric group has transitive action over $X=\{1, 2, 3, 4\}$?

score 188 · Answer 1 · 2020-04-08 23:53:52

How to prove that $S_4$ a symmetric group has transitive action over $X=\{1, 2, 3, 4\}$?

Published on 08 Apr 2020 - 23:53

score 36 · Answer 2 · 2020-04-09 23:14:13

Let $G$ be a group and let $S=G.$ Prove that the following $g\cdot x∶=gxg^{−1}$ for all $g \in G$ and $x \in S$ is an action of $G$ on $S$.

score 575 · Answer 3 · 2020-04-10 12:29:44

575

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Regular action vs. Transitive and Free

Published on 10 Apr 2020 - 12:29

score 211 · Answer 4 · 2020-04-10 18:52:43

211

Views

EXERCISE 4.1.2 of Abstract Algebra Dummit and Foote (3 Ed)

Published on 10 Apr 2020 - 18:52

score 39 · Answer 5 · 2020-04-10 19:45:57

39

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Connected components of Schreier Graph $Sch(G,S,X)$ are exactly the orbits of the action of G on X.

Published on 10 Apr 2020 - 19:45

score 117 · Answer 6 · 2020-04-12 21:03:59

117

Views

Clarificación en el ejemplo de la sección 4.3 Dummit and Foote (3Ed)

Published on 12 Apr 2020 - 21:03

score 47 · Answer 7 · 2020-04-13 01:45:56

Let $k(G)$ be the number of conjugation classes of the finite group $G$. Defining $C:=\{(x,y) \in G \times G: xy=yx\}$. Prove that $|C|=k(G)|G|$.

score 330 · Answer 8 · 2026-03-26 11:05:10

330

Views

Prove that $Z(S_n)=\{1\}$ for all $n\geq 3$.

Published on 26 Mar 2026 - 11:05

score 147 · Answer 9 · 2026-03-26 00:57:10

147

Views

About -derived subgroup- of a group $G$

Published on 26 Mar 2026 - 0:57

score 123 · Answer 10 · 2020-04-14 14:20:56

123

Views

Nontrivial continuous action $\mathbb{Z}_n \to S^1$ and quotient space

Published on 14 Apr 2020 - 14:20

score 92 · Answer 11 · 2026-03-25 22:23:51

92

Views

Semidirect product and group action

Published on 25 Mar 2026 - 22:23

score 46 · Answer 12 · 2020-04-16 00:34:31

46

Views

Prove that $\mathcal{O}_{(0,0)} = \{(0, 0)\}.$ The action is given by $\lambda \cdot (x, y) = (\lambda x, \lambda y).$

Published on 16 Apr 2020 - 0:34

score 128 · Answer 13 · 2020-04-16 19:43:30

128

Views

Two orbit equations: can them be used to prove some claim?

Published on 16 Apr 2020 - 19:43

score 190 · Answer 14 · 2020-04-17 02:05:44

190

Views

Cayley classic theorem: Every group $G$ is isomorphic to a subgroup of $\operatorname{Sym}(G)$.

Published on 17 Apr 2020 - 2:05

score 599 · Answer 15 · 2020-04-17 02:47:57

599

Views

Let $H$ be a subgroup of $G$, and suppose that $G$ acts by multiplication over the set $X:=G/H$ of the left-hand side classes of $H$ over $G$.

Published on 17 Apr 2020 - 2:47

List Question

How to prove that $S_4$ a symmetric group has transitive action over $X=\{1, 2, 3, 4\}$?

Let $G$ be a group and let $S=G.$ Prove that the following $g\cdot x∶=gxg^{−1}$ for all $g \in G$ and $x \in S$ is an action of $G$ on $S$.

Regular action vs. Transitive and Free

EXERCISE 4.1.2 of Abstract Algebra Dummit and Foote (3 Ed)

Connected components of Schreier Graph $Sch(G,S,X)$ are exactly the orbits of the action of G on X.

Clarificación en el ejemplo de la sección 4.3 Dummit and Foote (3Ed)

Let $k(G)$ be the number of conjugation classes of the finite group $G$. Defining $C:=\{(x,y) \in G \times G: xy=yx\}$. Prove that $|C|=k(G)|G|$.

Prove that $Z(S_n)=\{1\}$ for all $n\geq 3$.

About -derived subgroup- of a group $G$

Nontrivial continuous action $\mathbb{Z}_n \to S^1$ and quotient space

Semidirect product and group action

Prove that $\mathcal{O}_{(0,0)} = \{(0, 0)\}.$ The action is given by $\lambda \cdot (x, y) = (\lambda x, \lambda y).$

Two orbit equations: can them be used to prove some claim?

Cayley classic theorem: Every group $G$ is isomorphic to a subgroup of $\operatorname{Sym}(G)$.

Let $H$ be a subgroup of $G$, and suppose that $G$ acts by multiplication over the set $X:=G/H$ of the left-hand side classes of $H$ over $G$.

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