An automorphism is an isomorphism of a group to itself. An isomorphism is a 1-1 homomorphism. A homomorphism is a mapping which preserves group structure.
Let us work our way back up. A group is a set of elements and an operation
upon them; a mapping is simply that, a map from the set of group elements
to some other set, or in general a map of any set to another set, as long as
each element in the first set is mapped to only one element in the second.
One could map the group of the triangle Dih_{3},
to
quite legitimately, although there
would be little obvious reason to do so. ^{5}
Usually we map the group elements to
the set of elements of another group, and in particular we usually require
homomorphism: preserving group structure. The identity of one group should
map to the identity of the second group; pairs of inverses in the first group
should map to pairs of inverses in the second group; elements which commute
should map to elements which commute. This is captured by
Then an endomorphism is a homomorphism of a group to itself, e.g. mapping of Dih_{3} to and the other three elements to . An isomorphism is a homomorphism which is one-to-one, and an automorphism is a one-to-one isomorphism, e.g mapping to
generic | *s | homom. (into Cyc^{6}) | endomorphic | automorphic | *g | g^{-1}xg | |
I | cat | s | I | I | I | x | IxI |
s | dog | s^{2} | I | I | s | xs | s^{2}xs |
s^{2} | 4 | I | I | I | s^{2} | xs^{2} | sxs^{2} |
t | s | l | s^{3} | t | l | xt | txt |
l | s | r | s^{3} | t | r | xl | lxl |
r | 4 | t | s^{3} | t | t xr | rxr |