[tex] 0\rightarrow A\stackrel{\alpha}{\rightarrow}B\stackrel{\beta}{\rightarrow}C\rightarrow 0 [/tex]

in this short exact sequence, alpha has to be mono or injective and beta has to be epi or surjective. However, what i dont get is: given a sequence of some groups how does one test whether it is exact. In other words how does one construct function between groups which are either epi or mono? for example from Z2 to Z4 or others.

(this may seem a strange question for someone doing "algebraic ..." but i have not done any algebra beyond an introduction few years ago)

thank you