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Unformatted text preview: Math 913, Spring 2011 HW 1 Due Monday, 21 March 2011 1. Prove that a subgroup of the symmetric group Sym(), for finite, that is generated by a set S of transpositions (2cycles) is a direct sum L k i =1 Sym( i ) for pairwise disjoint subsets i of . Hint: On define a graph by letting a b if and only if ( a,b ) is one of the transpositions of S . Then the i will be the connected components of this graph. Remark. The result remains true for infinite with FSym( i ) in place of Sym( i ). 2. This problem will (among other things) show that the sign homomorphism on finite Sym() (and hence on arbitrary FSym()) is welldefined. Let F be a field and V the Sym() permutation module F with basis { e  } and action given by e .g = e .g . Let n 2 and set = { 1 , 2 ,...,n } so that Sym() = Sym( n ). ( a ) Recall that for g GL F ( V ), we have defined [ V,g ] = V ( g 1). Prove: ( i ) The subspace W of V is ginvariant with g trivial on V/W if and only if W [ V,g ]. ( ii ) [ V,gh ] [ V,g ] + [ V,h ]. ( b ) For g Sym( n ), let ( g ) be the smallest number of transpositions with product g . Prove that dim K [ V,g ] ( g ). ( c ) Prove that dim K [ V,g ] = ( g ) = n n k =  supp( g )   c k where n k is the number of cycles in g (including cycles of length 1) and c k is the number of cycles in g of length greater than 1. ( Hint: First prove that if g is a kcycle, then dim K [ V,g ] = ( g ) = k 1.) ( d ) For t a transposition, prove that ( gt ) = ( g ) 1. ( e ) Prove that sgn: g 7 ( 1) ( g ) = ( 1) dim k [ V,g ] is a homomorphism from Sym( n ) onto the multiplicative group 1. (This is the sign homomorphism.) 3. Let G = GL 2 ( R ), the group of 2 2 invertible matrices with entries from the ring with identity R . Set B = a b d a,d U ( R ) and t = 1 1 ....
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 Spring '11
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