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) If G is a group and X is a G-set, then the subset (g in G: for all x

) If G is a group and X is a G-set, then the subset (g in G: for all 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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Are the following statements correct? True False If G is a group and X is a G-set, then the subset {g in G: for all x in X one has gx = x} of Gis a normal subgroup of G. If G is a group, and X is a G-set, then for every x in X the stabiliser of x in Gis a normal subgroup of G. Let G be a finite group, and let X be a finite G-set such that for all non-identity elements g of G, g has no fixed points in x. Then #G divides #X. Let G be a finite group and X a G-set. Then the number of G- orbits in X divides #G.
Transcribed Image Text:Are the following statements correct? True False If G is a group and X is a G-set, then the subset {g in G: for all x in X one has gx = x} of Gis a normal subgroup of G. If G is a group, and X is a G-set, then for every x in X the stabiliser of x in Gis a normal subgroup of G. Let G be a finite group, and let X be a finite G-set such that for all non-identity elements g of G, g has no fixed points in x. Then #G divides #X. Let G be a finite group and X a G-set. Then the number of G- orbits in X divides #G.
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