Given G={3n:n element of Z} .
Closure: Let any a∈G,b∈G , so a=3n,b=3m where n∈Z,m∈Z .
Now a×b=3n×3m=3nm where nm∈Z ⟹ab∈G .
Associative: Let a,b,c∈G , so a=3n,b=3m,c=3k ,
hence (a×b)×c=(3nm)×3k=3nmk=3n×(3m×3k)=a×(b×c) .
Identity: For all a∈G , a=3m where m∈Z .
There exists e=1=30 where 0∈Z , such that a×e=e×a=a .
Inverse: For all a∈G , a=3m where m∈Z .
There exists b=3−m:a×b=b×a=1 .
Commutative: a×b=3nm=b×a ∀ a,b∈G .
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