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Let Sn be the group consisting of the set of all permutations on n letters, together with the operation of composition. Let Cn be the subset of Sn consisting of the identity and all cycles of length n. a) What is |Cn|? Be careful-the answer is not n + 1. b) Show that C3 is a subgroup of S3. c) Show that C4 is not a subgroup of S4. d) For which values of n is Cn a subgroup of Sn? Prove your answer.

Let Sn be the group consisting of the set of all permutations on n letters, together with the operation of composition. Let Cn be the subset of Sn consisting of the identity and all cycles of length n. a) What is |Cn|? Be careful-the answer is not n + 1. b) Show that C3 is a subgroup of S3. c) Show that C4 is not a subgroup of S4. d) For which values of n is Cn a subgroup of Sn? Prove your answer.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Let Sn be the group consisting of the set of all permutations on n letters, together with the operation of composition. Let Cn be the subset of Sn consisting of the identity and all cycles of length n. a) What is |Cn|? Be careful-the answer is not n + 1. b) Show that C3 is a subgroup of S3. c) Show that C4 is not a subgroup of S4. d) For which values of n is Cn a subgroup of Sn? Prove your answer.
Transcribed Image Text:Let Sn be the group consisting of the set of all permutations on n letters, together with the operation of composition. Let Cn be the subset of Sn consisting of the identity and all cycles of length n. a) What is |Cn|? Be careful-the answer is not n + 1. b) Show that C3 is a subgroup of S3. c) Show that C4 is not a subgroup of S4. d) For which values of n is Cn a subgroup of Sn? Prove your answer.
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