Adv Alegbra HW Solutions 60

Adv Alegbra HW Solutions 60 - 60 2.81 Dene W = (1 2)(3 4) ,...

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60 2.81 De f ne W =h ( 12 )( 34 ) i , the cyclic subgroup of S 4 generated by ( )( ) . Show that W is a normal subgroup of V , but that W is not a normal sub- group of S 4 . Conclude that normality is not transitive: K C H and H C G need not imply K C G . Solution. Since every subgroup of an abelian group is a normal subgroup, W is a normal subgroup of V .However , W is not a normal subgroup of S 4 , for conjugating ( )( ) by ( 13 ) gives ( )( )( )( ) = ( 23 )( 14 )/ W . 2.82 Let G be a f nite group written multiplicatively. Prove that if | G | is odd, then every x G has a square root. Conclude, using Exercise 2.45, that there exists exactly one g G with g 2 = x . Solution. The function ϕ : G G ,de f ned by ϕ( g ) = g 2 , is a homo- morphism because G is abelian. Now ker ϕ ={ g G : g 2 = 1 }={ 1 } because G has odd order, hence has no elements of order 2. It follows that ϕ is injective. By Exercise 2.13, the function ϕ is surjective; that is, for each x G , there is g G with x = g ) = g 2 . We have shown that x has a square root. This square root is unique, for if g 2 = x = h 2 for some h
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.

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