[Math] Definition of Simple Group

Question

Herstein defined the definition of a simple group as follows:

A group is said to be simple if it has no non-trivial homomorphic image.

Please help me to understand what is meant by non-trivial homomorphic image.

Thanks.

Answer 1

Put another way, any homomorphism from the group is a monomorphism (so the image of the homomorphism is the group we started with, for all intents and purposes) or the homomorphism mapping everything to the identity (so the image of the group is the trivial subgroup generated by the identity of the group we're mapping to). If the homomorphism isn't injective and isn't the homomorphism mapping everything to the identity, then the group's image under the homomorphism doesn't "look like" the group, nor like the trivial subgroup. That's what Herstein means by "non-trivial homomorphic image."

Equivalently, the kernel of any homomorphism from the group is the trivial subgroup generated by the group's identity element, or it is the whole group. Since for any normal subgroup there is a homomorphism having it as a kernel (projection onto quotient group), and any homomorphism's kernel is a normal subgroup of the domain, then Herstein's definition is equivalent to: "A group is said to be simple if it has no non-trivial proper normal subgroups."