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this problem, consider the group G = S3 acting on X = S4 by conjugation. other words, given an element gE S3 (thought of as a permutation of {1,2,3}) d an element x E S4 (thought of as a permutation of {1, 2, 3, 4}), the action is efined by g •x = gxg¯'. or example, (12) · (34) = (12)(34)(12)-1 = (34). nswer the following, being sure to carefully justify your answers. a) Describe the orbit Orbg((14)), and the stabiliser Stabg((14)). b) Is the action faithful? c) How many distinct orbits are there in X?

this problem, consider the group G = S3 acting on X = S4 by conjugation. other words, given an element gE S3 (thought of as a permutation of {1,2,3}) d an element x E S4 (thought of as a permutation of {1, 2, 3, 4}), the action is efined by g •x = gxg¯'. or example, (12) · (34) = (12)(34)(12)-1 = (34). nswer the following, being sure to carefully justify your answers. a) Describe the orbit Orbg((14)), and the stabiliser Stabg((14)). b) Is the action faithful? c) How many distinct orbits are there in X?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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In this problem, consider the group G = S3 acting on X = S4 by conjugation. In other words, given an element g E S3 (thought of as a permutation of {1, 2, 3}) and an element x E S4 (thought of as a permutation of {1, 2, 3, 4}), the action is defined by g •x = gxg¬!. For example, (12) · (34) = (12)(34)(12)-1 = (34). Answer the following, being sure to carefully justify your answers. (a) Describe the orbit Orbg((14)), and the stabiliser Stabg((14)). (b) Is the action faithful? (c) How many distinct orbits are there in X?
Transcribed Image Text:In this problem, consider the group G = S3 acting on X = S4 by conjugation. In other words, given an element g E S3 (thought of as a permutation of {1, 2, 3}) and an element x E S4 (thought of as a permutation of {1, 2, 3, 4}), the action is defined by g •x = gxg¬!. For example, (12) · (34) = (12)(34)(12)-1 = (34). Answer the following, being sure to carefully justify your answers. (a) Describe the orbit Orbg((14)), and the stabiliser Stabg((14)). (b) Is the action faithful? (c) How many distinct orbits are there in X?
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