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Unformatted text preview: Math 5125 Monday, September 5 Second Homework Solutions 1. Let G be a group with distinct normal subgroups A , B such that  G : A  =  G : B  = 2. (a) Prove that AB = G . (b) Prove that A / A ∩ B and B / A ∩ B are distinct normal subgroups of G / A ∩ B of order 2 and index 2. (c) Deduce that G has at least 3 subgroups of index 2. (a) AB is a subgroup of G which strictly contains A . Since A has index 2 in G , it follows that AB has index 1 in G and so AB = G . (b) Note that A ∩ B is a normal subgroup of G because A and B are normal subgroups. By the subgroup correspondence theorem, A / A ∩ B and B / A ∩ B are normal sub groups of G / A ∩ B , and they are distinct because A and B are distinct. Since A and B have index 2 in G , we see that A / A ∩ B and B / A ∩ B have index 2 in G / A ∩ B . Finally A / A ∩ B ∼ = AB / B ∼ = G / B , which shows that A / A ∩ B has order 2, and similarly B / A ∩ B has order 2....
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 Fall '07
 PALinnell
 Math, Algebra, Normal subgroup, Subgroup, normal subgroups, subgroup correspondence theorem

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