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Adv Alegbra HW Solutions 54

# Adv Alegbra HW Solutions 54 - 54 2.54(i Dene the special...

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54 2.54 (i) De fi ne the special linear group by SL ( 2 , R ) = { A GL ( 2 , R ) : det ( A ) = 1 } . Prove that SL ( 2 , R ) is a subgroup of GL ( 2 , R ) . Solution. It suf fi ces to show that if A , B SL ( 2 , R ) , then so is AB 1 ; that is, if det ( A ) = 1 = det ( B ) , then det ( AB 1 ) = 1. Since det ( UV ) = det ( U ) det ( V ) , it follows that 1 = det ( E ) = det ( BB 1 ) = det ( B ) det ( B 1 ). Hence, det ( AB 1 ) = det ( A ) det ( B 1 ) = 1. (ii) Prove that GL ( 2 , Q ) is a subgroup of GL ( 2 , R ) . Solution. Both the product of two matrices with rational entries and the inverse of a matrix with rational entries have rational en- tries. 2.55 Give an example of two subgroups H and K of a group G whose union H K is not a subgroup of G . Solution. If G = S 3 , H = ( 1 2 ) , and K = ( 1 3 ) , then H K is not a subgroup of G , for ( 1 2 )( 1 3 ) = ( 1 3 2 ) / H K . 2.56 Let G be a fi nite group with subgroups H and K . If H K , prove that [ G : H ] = [ G : K ][ K : H ] . Solution. If G is a fi nite group with subgroup H , then [ G : H ] = | G | / | H | . Hence, if H K , then [ G : K ][ K : H ] = ( | G | / | K | ) · ( | K | / | H | ) = | G | / | H | = [ G : H ] . 2.57 If H and K are subgroups of a group G and if | H | and | K | are relatively prime, prove that H K = { 1 } . Solution. By Lagrange s theorem, | H K | is a divisor of
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