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4. Let G be a group with 3-5-17 elements. As usual, let N, denote the number of p-Sylow subgroups with p elements (for p = 3,5, 17). Let A, B, C be 3-Sylow, 5-Sylow, and 17-Sylow subgroups, respectively. (a) (b) Use Sylow's Theorem to deduce information about N3, N5, N7. Explain why C is a normal subgroup of G. 1 (c) Explain why either A or B has to be a normal subgroup of G. (Try counting the possible number of elements of order 3 and of order 5 in G.) (d) Explain why AB is a subgroup of G. (e) Explain why AB is cyclic.

4. Let G be a group with 3-5-17 elements. As usual, let N, denote the number of p-Sylow subgroups with p elements (for p = 3,5, 17). Let A, B, C be 3-Sylow, 5-Sylow, and 17-Sylow subgroups, respectively. (a) (b) Use Sylow's Theorem to deduce information about N3, N5, N7. Explain why C is a normal subgroup of G. 1 (c) Explain why either A or B has to be a normal subgroup of G. (Try counting the possible number of elements of order 3 and of order 5 in G.) (d) Explain why AB is a subgroup of G. (e) Explain why AB is cyclic.

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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4. Let G be a group with 3-5-17 elements. As usual, let Np denote the number of p-Sylow subgroups with p elements (for p = 3,5, 17). Let A, B, C be 3-Sylow, 5-Sylow, and 17-Sylow subgroups, respectively. 2 (a) (b) Use Sylow's Theorem to deduce information about N3, N5, N7. Explain why C is a normal subgroup of G. 1 (c) Explain why either A or B has to be a normal subgroup of G. (Try counting the possible number of elements of order 3 and of order 5 in G.) (d) Explain why AB is a subgroup of G. Explain why AB is cyclic. It turns out that G is the internal direct product of AB and C. As a bonus, feel free to give an explanation, if you see it!
Transcribed Image Text:4. Let G be a group with 3-5-17 elements. As usual, let Np denote the number of p-Sylow subgroups with p elements (for p = 3,5, 17). Let A, B, C be 3-Sylow, 5-Sylow, and 17-Sylow subgroups, respectively. 2 (a) (b) Use Sylow's Theorem to deduce information about N3, N5, N7. Explain why C is a normal subgroup of G. 1 (c) Explain why either A or B has to be a normal subgroup of G. (Try counting the possible number of elements of order 3 and of order 5 in G.) (d) Explain why AB is a subgroup of G. Explain why AB is cyclic. It turns out that G is the internal direct product of AB and C. As a bonus, feel free to give an explanation, if you see it!
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