# algebra exercises

May 29th, 2015
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Every subgroup of an abelian group is normal.If G is an abelian group, then every subgroup of G is normal.Prove: If H is a subgroup of the abelian group G and g ∈ G and h ∈ H, then ghg-1= hgg-1 commutative for abelian group

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Prove the following : 1. Every subgroup of an abelian group is normal.If G is an abelian group, then every subgroup of G is normal.Prove:If H is a subgroup of the abelian group G and g G and h H, then ghg-1= hgg-1 commutative for abelian group = he = h h H.2. Prove the center of a group GZ ( G) = { a G | ag = ga, G }is a normal subgroup of G. Want to show the center Z ( G) of a group G is a normal subgroup of G i. Z ( G) is nonemptyIs e in element of Z ( G) ?Yes, G .Since eg = ge ii. Closed under operation.Since a,bTake : Want to show LHSG is a group associative.= = G is associative.= = G is associative.RHSG is a group associative.= = G is associative.= = G is associative.Since, , so is associative.iii. Closed under inverse since Take and a

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