creating an example map

[math_stresser]

Create an example map from one finite group to another finite group that is bijective but not a homomorphism. Explicitly demonstate how your map fails to be a homomorphism.

I have been having the hardest time with this problem. I keep thinking I get an answer, only to realize that it's homomorphic. Or that it's not homomorphic, but it's also not bijective.

I finally came up with an example, but it was just one random domain to another random codomain. Then I realized it needed to be one finite group to another.

Some of the examples I have tried are y=2x. It is not injective. Then I tried just using positive numbers, but it's still not a finite group. When I use just a small group: i.e. -1,0, and 1, that doesn't work because it doesn't have closure.

I really am struggling here, so even just a suggestion would be GREATLY appreciated!

 

[daon]

There are a few properties a homomorphism has to satisfy. One of them is that \(\displaystyle f(e_G)=e_H\). Any mapping that breaks this rule will fail to be a homomorphism.

For an easy example, try a map from \(\displaystyle (Z_3,+)\) to itself in which 0 is not sent to zero.