Question #133606
Let G={3^n;n element of Z} prove that (G,•)
is an abelian group
1
Expert's answer
2020-09-24T18:07:51-0400

Given G={3n:n element of Z}G=\{3^n : n \ element \ of \ \Z\} .

Closure: Let any aG,bGa\in G, b\in G , so a=3n,b=3ma = 3^n, b = 3^m where nZ,mZn \in \Z, m\in \Z .

Now a×b=3n×3m=3nma \times b = 3^n \times 3^m = 3^{n m} where nmZn m \in \Z     abG\implies ab \in G .

Associative: Let a,b,cGa, b, c \in G , so a=3n,b=3m,c=3ka = 3^n, b = 3^m, c = 3^k ,

hence (a×b)×c=(3nm)×3k=3nmk=3n×(3m×3k)=a×(b×c)(a \times b) \times c = (3^{nm}) \times 3^k = 3^{nmk} = 3^n \times (3^m \times 3^k) = a \times (b\times c) .

Identity: For all aGa \in G , a=3ma = 3^m where mZm \in \Z .

There exists e=1=30e = 1 = 3^0 where 0Z0 \in \Z , such that a×e=e×a=aa \times e = e\times a = a .

Inverse: For all aGa \in G , a=3ma = 3^m where mZm \in \Z .

There exists b=3m:a×b=b×a=1b = 3^{-m} : a\times b = b\times a = 1 .

Commutative: a×b=3nm=b×a   a,bGa \times b = 3^{nm} = b \times a \ \ \forall \ a, b \in G .


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