A sequence of groups is just a list of groups with homomorphisms
going down the list: . We’ll use to refer to the homomorphism
from to . We say that a sequence is “exact” if the image of is
the kernel of for each . What does this mean? First off, if we
start with an element of and hit it with we get something in .
Since this is the same as , if we now apply we go to the
identity element of . That is, the composition is always the
trivial homomorphism sending all of to the identity in . On the
other hand, if we have an element in the kernel of it’s also in
the image of . This means that if an element gets sent to the
identity in the next group, it must have come from an element in
the previous group. So why should we care? Well, a number of
different things can be very nicely said with exact sequences.
If we write for the group containing only one element, we can
set up a sequence: . What does it mean for this sequence to be
exact? Well there’s only one homomorphism from to any group, and
its image is just the