Homework8 - Homework 8 Solutions 1. (a) False. e.g. S3 has...

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Homework 8 Solutions 1. (a) False. e.g. S 3 has 9 Sylow 2-subgroups. (b) True. G abelian implies all subgroups are normal, which implies that each Sylow sub- group is unique. (c) True. Sylow’s Theorem part 1. (d) True. If y = gx then G y = gG x g - 1 . (e) False. e.g. G = S 3 and H = h (12) i . N G ( H ) = H 6 C G . (f) True. Definition of normalizer. (g) False. If the Sylow p -subgroup isn’t normal then this doesn’t hold. (h) True. The size of the Sylow p -subgroup is p n where n is the largest power of p appearing in the order of the group. Since | G | = | H | , n is the same for both of them. (i) False. 8 6≡ 1 (mod 5). 2. Let G = S 3 × S 3 . Subgroups of size 4 are the Sylow 2-subgroups. Let P = h (12) i × h (12) i , Q = h (13) i × h (13) i and R = h (12) i × h (13) i . P , Q and R are all Sylow 2-subgroups. P Q = { ((1) , (1)) } = { e } but P R = { e, ((12) , (1)) } . 3. (a) 300 = 2
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This note was uploaded on 04/19/2011 for the course MATH 21373 taught by Professor Johnson during the Spring '11 term at Carnegie Mellon.

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Homework8 - Homework 8 Solutions 1. (a) False. e.g. S3 has...

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