VW - CVM 1.1

# Group Homomorphisms

When we set out to construct recipes for functions with negating symmetries, we started with the group G, the actual symmetries of a pattern, and extended it to E . This is most natural when attempting to construct functions of a given pattern.

Group homomorphisms are useful for discussing the alternative, starting with E, the group of extended symmetries, and reducing it to G. This approach seems most useful in the algebraic proof that all patterns have been identified. For convenience, we let {1,-1} indicate the cyclic group of order 2, using multiplicative notation. (See the informal discussion of anti-symmetries for more information ).

The following proposition, whose proof is evident , gives an elegant way to describe every situation where symmetries or anti-symmetries are present in a wallpaper function.

Proposition: Suppose E is any of the 17 wallpaper groups, P is any homomorphism of E to {1,-1}, and f satisfies f(h x) = P(h) f(x) for every x in R2 and h in E. Let G be the kernel of E; that is P-1(1). Let k be any element of E with P(k) = -1. Then f is G-invariant and k-negating.

Conversely, if f is G-invariant and k-negating, then the equation above can be used to define a homomorphism P with G as its kernel.

From this we see that G is normal in E, as the kernel of P.

It seems, then, that counting the ways in which a function could be G-invariant and k-negating amounts to counting the homomorphisms of E to {1,-1}. But this is not quite correct; a notion of equivalence is needed, as we see in the next section.

Communications in Visual Mathematics, vol 1, no 1, August 1998.