Suppose $G$ is a nontrivial group, not necessarilly countable, and suppose we have two surjective group homomorphisms $\alpha, \beta: G\rightarrow G$, then we could form the ''amalgamated free product'' $H=G*_{G}G$ by using these two maps, see the precise definition of "amalgamated free product" here: http://en.wikipedia.org/wiki/Free_product#Generalization:_Free_product_with_amalgamation
then, is H always a nontrivial group? If not, then what else assumptions needed to make it is nontrivial.