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Let (G,G,\ast) be the following group. The set GG is {0,1}×\times{0,1,2}; that is

G={(0,0),(0,1),(0,2),(1,0),(1,1),(1,2)}G=\{(0,0),(0,1),(0,2),(1,0),(1,1),(1,2)\}

The operation \ast is defined by

(a,b)(c,d)=(a+c mod 2,b+d mod 3)(a,b)\ast (c,d)=(a+c\text{ mod }2, b+d\text{ mod }3)

For example, (1,2)\ast(1,2)=(0,1). Find an isomorphism from (G,)(G,\ast) to (Z6,)(\mathbb{Z}_6,\oplus).

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DEFINITIONS

Group isomorphism\textbf{Group isomorphism} f:GHf:G\rightarrow H is called a group isomorphism if ff is one-to-one and onto and g,hG,f(gh)=f(g)f(h)\forall g,h\in G, f(g\ast h)=f(g)\star f(h).

The function ff is onto\textbf{onto} if and only if for every element bBb\in B there exist an element aAa\in A such that f(a)=bf(a)=b.

The function ff is one-to-one\textbf{one-to-one} if and only if f(a)=f(b)f(a)=f(b) implies that a=ba=b for all aa and bb in the domain.

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