s09_mthsc851_hw02

s09_mthsc851_hw02 - G is the disjoint union of its ( A,B...

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MTHSC 851 (Abstract Algebra) Dr. Matthew Macauley HW 2 Due wednesday Jan. 28, 2009 (1) Suppose G is finite, p is the smallest prime dividing | G | , H G , and [ G : H ] = p . Show that H C G . (2) Suppose [ G : H ] is finite. Show that there is a normal subgroup K of G with K H , such that [ G : K ] is finite. (3) Suppose H S n but H 6≤ A n . Show that [ H : A n H ] = 2. (4) Prove that if H and K are normal subgroups of a group G and HK = G then G/ ( H K ) = ( G/H ) × ( G/K ) . (5) Prove the tower law : If K H G , then [ G : K ] = [ G : H ][ H : K ]. (6) Suppose G is finite, H G , [ G : H ] = n , and | G | 6| n !. Show that there is a normal subgroup K of G , K 6 = 1, such that K H . (7) If | G | = p n for some prime p , and 1 6 = H C G , show that H Z ( G ) 6 = 1. (8) If A,B G and y G define ( A,B ) -double coset AyB = { ayb | a A, b B } . Show that
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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 ) x-1 . (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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This note was uploaded on 03/11/2012 for the course MTHSC 851 taught by Professor Staff during the Fall '08 term at Clemson.

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