2. Let G be a group of order #G = p'm where r> 0 is an integer, p is a prime number, and p does not divide m. Let S C G be a p-Sylow subgroup of G and let X = G/S is the set of left cosets. (a) Show that p does not divide #X. (b) Let H be a subgroup of G. Let H act on X by h.gS = (hg)S for h = H, gS E X. i. Show that if h E H is such that the left cosets gS and hgS are equal, then there is s ES such that h = gsg-¹ ii. Deduce that the stabilizer Has = gSg-¹nH.
2. Let G be a group of order #G = p'm where r> 0 is an integer, p is a prime number, and p does not divide m. Let S C G be a p-Sylow subgroup of G and let X = G/S is the set of left cosets. (a) Show that p does not divide #X. (b) Let H be a subgroup of G. Let H act on X by h.gS = (hg)S for h = H, gS E X. i. Show that if h E H is such that the left cosets gS and hgS are equal, then there is s ES such that h = gsg-¹ ii. Deduce that the stabilizer Has = gSg-¹nH.
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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only answer b, where gS = hgS
Transcribed Image Text:2. Let G be a group of order #G = p^m where r > 0 is an integer, p is a prime
number, and p does not divide m. Let S C G be a p-Sylow subgroup of G
and let X = G/S is the set of left cosets.
(a) Show that p does not divide #X.
(b) Let H be a subgroup of G. Let H act on X by h.gS = (hg)S for h = H,
gS E X.
i. Show that if h E H is such that the left cosets gS and hgS are equal,
then there is s ES such that h = gsg-¹
-1
ii. Deduce that the stabilizer Has = gSg-¹nH.
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