Let G be a group of order 24. Suppose that the center of G has order two and that G/Z(G) is isomorphic to the alternating group A4. a) Show that G has a unique Sylow 2-subgroup P. b) Let H be a Sylow 3-Subgroup of G. Show that G=HP c) Show that the order of Aut(P), the automorphism group of P, is divisible by 3

Let G be a group of order 24. Suppose that the center of G has order two and that G/Z(G) is isomorphic to the alternating group A4. a) Show that G has a unique Sylow 2-subgroup P. b) Let H be a Sylow 3-Subgroup of G. Show that G=HP c) Show that the order of Aut(P), the automorphism group of P, is divisible by 3

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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Question

Let G be a group of order 24. Suppose that the center of G has order two and that G/Z(G) is isomorphic to the alternating group A4.

a) Show that G has a unique Sylow 2-subgroup P.

b) Let H be a Sylow 3-Subgroup of G. Show that G=HP

c) Show that the order of Aut(P), the automorphism group of P, is divisible by 3

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