## Related questions with answers

Question

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

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

The operation $\ast$ is defined by

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

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

Solution

VerifiedStep 1

1 of 3DEFINITIONS

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

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

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

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