Explain why the correspondence \$x mapsto 3x\$ from \$Bbb Z_12\$ to \$Bbb Z_10\$ is not a homomorphism.

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Here photo of \$1\$ is \$3\$ and also \$|3| = 10 \$ which does not divide \$12\$, the order of \$1\$. So this can"t be homomorphism.

Is this correct?

This "map" is doomed from the outcollection, as it is not well defined.

As it fails to be a bonified attribute, it absolutely can"t be a homomorphism.

Besides, it is straightforward to check out the just homomorphism, various other than the trivial one, from \$Bbb Z_12\$ to \$Bbb Z_10\$ is \$h\$, offered by \$h(1)=5\$.

Call your map \$f\$. On the one hand also \$f(0)= f(12cdot 1)\$. Thus, for \$f\$ to be a homomorphism, one need to also have actually \$\$ 0 = f( 0 ) = f(12cdot 1) = 12 cdot f(1) .\$\$So (a variable of) \$12\$ would certainly need to kill \$f(1)\$ in \$steustatiushistory.orgbb Z_10\$ - however \$f(1)=3\$, and also \$\$ 12 cdot 3 = 6 ot= 0in steustatiushistory.orgbb Z_10.\$\$So no element of \$12\$ kills \$f(1)\$.

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