Mappings and 'Morphisms

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 Dih₃, $\{I, s, s^2, t, l, r\}$ to $\{cat, dog, ice cream, monkey, 4, 2\}$ 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

$$\phi(a)\phi(b) = \phi(ab)$$

where $\phi$ is the mapping under consideration. We can see that this does express the conditions above:

$$\phi(x) = \phi(Ix) = \phi(I)\phi(x)$$

so $\phi(I)$ must be the identity of the second group. And

$$\phi(I) = \phi(xx^{-1}) = \phi(x)\phi(x^{-1})$$

so $\phi(x^{-1})$ must equal $\phi(x)^{-1}$. Finally

ab = ba

f(ab) = f(ba)

f(a)f(b) = f(b)f(a)

Then an endomorphism is a homomorphism of a group to itself, e.g. mapping $\{I, s, s^2\}$ of Dih₃ to $\{I\}$ and the other three elements to $\{t\}$. An isomorphism is a homomorphism which is one-to-one, and an automorphism is a one-to-one isomorphism, e.g mapping $\{I, s, s^2, t, l, r\}$ to $\{I, s^2, s, t, r, l\}$

Sample mappings

	generic	*s	homom. (into Cyc6)	endomorphic	automorphic	*g	g-1xg
I	cat	s	I	I	I	x	IxI
s	dog	s2	I	I	s	xs	s2xs
s2	4	I	I	I	s2	xs2	sxs2
t	s	l	s3	t	l	xt	txt
l	s	r	s3	t	r	xl	lxl
r	4	t	s3	t	t xr	rxr