Engineering Mathematics2013Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917)Page 7Small Problems1)Let2,4,6,8E. Show that,Eand,Eare semi–groups but not monoids.2)Prove that the identity element of a group G is unique.3)Prove that the inverse element of a group G is unique.4)State and prove the cancellation law in a group.5)IfaG, G be a group , then prove that11aa6)In a group G,111**,abbaa bG7)Find the multiplication inverse of each element in11Z.8)If every element in a group is its own inverse then the group must be abelian.9)For anyaG,2aethen prove that G is an abelian .10)Prove that a group can not have any element which is idempotent except the identityelement.11)Prove that a group G is abelian if and only if222**,ababa bG12)Find all cosets of a sub groups21,Haof a group231, ,,Ga aaunder usualmultiplication, where41a13)Let1,1, ,Giibe a group and1,1Hbe a sub group of G.What is thenumber of distinct cosets of H in G.14)Prove that a subgroup of an abelian group is a normal subgroup.Theorems1)Prove that if G is an abelian group then for all,a bGand all integer n,**nnnabba.2)If S = N X Nthe set of ordered pairs of positive integers with the operation * defined by,*,,a bc dadbc bdand if:,*,fSQis defined by,/fa bab , then show thatfis a semi–group homomorphism.3)IfSis the set of all ordered pairs,a bof real numbers wit the binary operationdefined by,,,a bc dac bd, where, , ,a b c dare real, prove that,Sis a commutative group.4)Show that the set of all positive rational numbers forms an abelian group under thecomposition defined by*2abab5)On the set Q of all rational numbers, the operation * is defined by*abababShow that, under the operation *, Q is a commutative monoid.......