List questions from permutation-cycles

Published on 11 Apr 2026 - 21:10

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

Published on 13 Apr 2026 - 23:10

27

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

Published on 17 Apr 2026 - 14:43

133

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Permutations and terminology

Published on 10 May 2026 - 9:31

354

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

Published on 09 Apr 2026 - 3:55

57

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Order and sign of a cycle

Published on 10 May 2026 - 9:38

289

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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 17 Apr 2026 - 11:48

71

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

Published on 15 Apr 2026 - 23:29

202

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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 15 Apr 2026 - 0:06

74

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Alternate definition of symmetric group action

Published on 10 Apr 2026 - 11:01

726

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Explanation of an unexpected observation in modular arithmetic

Published on 08 Apr 2026 - 16:27

49

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commutativity of cycle permutations

Published on 12 Apr 2026 - 2:13

9.1k

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Number of homomorphisms between two arbitrary groups

Published on 14 Apr 2026 - 12:46

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 14 Apr 2026 - 13:00

659

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Proving the sign map is a homomorphism

Published on 16 Apr 2026 - 0:51

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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