List questions from symmetric-groups

Published on 06 Apr 2020 - 18:11

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Reference for ${\rm sgn} ( \sigma \tau ) = {\rm sgn} ( \sigma ) {\rm sgn} ( \tau )$ and ${\rm sgn} ( \sigma ) ={\rm sgn} ({ \sigma}^{-1} )$

Published on 07 Apr 2020 - 12:41

39

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How to find the number of left coset of $H$ in symmetric group $S_7$

Published on 26 Mar 2026 - 11:01

39

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There exists a group element $\sigma\in S_7$ under some conditions?

Published on 26 Mar 2026 - 11:00

330

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

Published on 26 Mar 2026 - 11:05

71

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Does $\{a,b,c,c^2\}$ generate the same group as $\{a,b,c\}$?

Published on 26 Mar 2026 - 16:10

50

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Show that $\operatorname{sign}\sigma=(-1)^{pq}$ for a particular $\sigma\in S(p+q)$

Published on 20 Apr 2020 - 12:32

494

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General Cayley's theorem: Let $G$ be a group and $H$ a subgroup of $G$ such that $[G:H]=n$.

Published on 20 Apr 2020 - 19:09

427

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Symmetry group of a line segment in Euclidean spaces

Published on 27 Mar 2026 - 8:42

247

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Homomorphic image of an alternating group

Published on 28 Mar 2026 - 10:55

393

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Orders of Elements in Symmetric Groups

Published on 22 Apr 2020 - 17:13

235

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Prove for $\sigma \in S_n$, $\sigma(a_1, … , a_n)\sigma^{-1}= (\sigma(a_1),…, \sigma(a_n))$

Published on 26 Mar 2026 - 12:37

133

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Proof that given $p$ prime and $n \in \mathbb{N}$, $n!$ divides $(p^n - 1)(p^n - p)\cdots(p^n - p^{n-1})$.

Published on 25 Mar 2026 - 8:10

82

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Orbits of pairs of multi-indices under the diagonal action of the symmetric group

Published on 25 Mar 2026 - 12:50

118

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How to determine the smallest $\ m\ $ with $\ G\le S_m\ $?

Published on 27 Apr 2020 - 12:47

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A cycle is an even permutation iff its length is odd

Reference for ${\rm sgn} ( \sigma \tau ) = {\rm sgn} ( \sigma ) {\rm sgn} ( \tau )$ and ${\rm sgn} ( \sigma ) ={\rm sgn} ({ \sigma}^{-1} )$

How to find the number of left coset of $H$ in symmetric group $S_7$

There exists a group element $\sigma\in S_7$ under some conditions?

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

Does $\{a,b,c,c^2\}$ generate the same group as $\{a,b,c\}$?

Show that $\operatorname{sign}\sigma=(-1)^{pq}$ for a particular $\sigma\in S(p+q)$

General Cayley's theorem: Let $G$ be a group and $H$ a subgroup of $G$ such that $[G:H]=n$.

Symmetry group of a line segment in Euclidean spaces

Homomorphic image of an alternating group

Orders of Elements in Symmetric Groups

Prove for $\sigma \in S_n$, $\sigma(a_1, … , a_n)\sigma^{-1}= (\sigma(a_1),…, \sigma(a_n))$

Proof that given $p$ prime and $n \in \mathbb{N}$, $n!$ divides $(p^n - 1)(p^n - p)\cdots(p^n - p^{n-1})$.

Orbits of pairs of multi-indices under the diagonal action of the symmetric group

How to determine the smallest $\ m\ $ with $\ G\le S_m\ $?

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