Unformatted text preview: G is the disjoint union of its ( A,B )double cosets. Show that  AyB  = [ A y : A y ∩ B ] ·  B  if A and B are ﬁnite. (9) Let G be a group of order 15, which acts on a set S with 7 elements. Show the group action has a ﬁxed point. (10) Suppose G acts on S , x ∈ G , and x ∈ S . Show that Stab G ( xs ) = x Stab G ( s ) x1 . (11) Prove that if G contains no subgroup of index 2, then any subgroup of index 3 is normal in G . (12) Suppose that H and K both have ﬁnite index in G . Prove that [ G : H ∩ K ] ≤ [ G : H ][ G : K ]....
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 Fall '08
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 Algebra, Group Theory, Normal subgroup, Subgroup, Dr. Matthew Macauley, Suppose H Sn

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