List questions from permutation-cycles

Published on 29 Nov 2018 - 12:18

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Express (1,...,n) as a product of 2-cycles

Published on 30 Nov 2018 - 12:17

24

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find all $x \in S_5$ such that $x^3 = (12)$

Published on 01 Dec 2018 - 13:30

128

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Permutations and terminology

Published on 02 Dec 2018 - 19:09

353

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Cycle structure of the permutation $x \mapsto p·x\operatorname{mod}q$ for coprime $p,q$

Published on 23 Feb 2026 - 1:08

52

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Order and sign of a cycle

Published on 05 Dec 2018 - 18:02

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Compute all $X_{g}$ and all $G_{x}$ for $X = \left \{1, 2, 3\right \}$, $G = S_{3} = \left \{(1), (12), (13), (23), (123), (132)\right \}$.

Published on 05 Dec 2018 - 18:56

68

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Find the conjugacy class $O_{(234)}$ if $G=A_{4}$ is acting on itself by conjugation.

Published on 05 Dec 2018 - 19:21

199

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Let $n$ be an odd positive integer and $a\in S_n$ be an $n$-cycle. Show that the order of $C(a)$ must be odd.

Published on 08 Dec 2018 - 22:23

71

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Alternate definition of symmetric group action

Published on 10 Dec 2018 - 18:30

723

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Explanation of an unexpected observation in modular arithmetic

Published on 11 Dec 2018 - 22:59

46

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commutativity of cycle permutations

Published on 17 Dec 2018 - 13:37

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Number of homomorphisms between two arbitrary groups

Published on 20 Dec 2018 - 18:23

1.3k

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The centralizer of a cycle is the group generated by the elements of $S_n$ that are disjoint from the cycle and the group generated by the cycle.

Published on 24 Dec 2018 - 17:23

656

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Proving the sign map is a homomorphism

Published on 26 Dec 2018 - 17:21

List Question

Definition of a permutation as product of cycles

Express (1,...,n) as a product of 2-cycles

find all $x \in S_5$ such that $x^3 = (12)$

Permutations and terminology

Cycle structure of the permutation $x \mapsto p·x\operatorname{mod}q$ for coprime $p,q$

Order and sign of a cycle

Compute all $X_{g}$ and all $G_{x}$ for $X = \left \{1, 2, 3\right \}$, $G = S_{3} = \left \{(1), (12), (13), (23), (123), (132)\right \}$.

Find the conjugacy class $O_{(234)}$ if $G=A_{4}$ is acting on itself by conjugation.

Let $n$ be an odd positive integer and $a\in S_n$ be an $n$-cycle. Show that the order of $C(a)$ must be odd.

Alternate definition of symmetric group action

Explanation of an unexpected observation in modular arithmetic

commutativity of cycle permutations

Number of homomorphisms between two arbitrary groups

The centralizer of a cycle is the group generated by the elements of $S_n$ that are disjoint from the cycle and the group generated by the cycle.

Proving the sign map is a homomorphism

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