Something that has confused me for long is the condition for injectivity for a homomorphism. The condition is that the kernel should just be the identity element.
I used to think that maybe this condition for injectivity applies to all mappings, and wondered why I hadn’t come across this earlier.
No. This condition applies only to homomorphisms due to their special property:
. This condition would no longer be valid if the mapping was so defined:
, where
.