TITLE: Definition of Simple Group
QUESTION [7 upvotes]: 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.

REPLY [3 votes]: The word trivial is used in mathematics in very context-dependent ways.  Here non-trivial homomorphic image is understood to exclude two trivial cases that occur for every group.
A group $G$ has itself (or an isomorphic copy) as a homomorphic image, and also the image of a trivial group containing only one element (the trivial identity group).  These two homomorphic images are considered trivial cases, and the definition of a simple group amounts to saying there are no other possibilities for a homomorphic image.
As other Answers point out, the definition of simple group is often stated as an equivalent property on normal subgroups, i.e. that there are only the group $G$ itself and the trivial (identity) subgroup which are normal in $G$.  These forms of definition are equivalent by the First Isomorphism Theorem (for groups).