VW - CVM 1.1

The Group pmm

	Generators:		t1, t2, R, M
	Relations:		M2 = 0, R2 = 0
	Cell diagram:

This table is the most tedious, as sixteen possible homomorphisms reduce to only six equivalence classes. To reduce the size of the table, we indicate with parentheses whenever an equivalent pattern would result when t₂ is substituted for t₁.

Table: Six homomorphisms from pmm

type	P(k) = -1	wallpaper type of kernel group, with remarks
pmm	none	pmm, same as the extended group
pm'm'	M	p2, all mirrors negating
pmm'	R	pm = {t1, t2, M}, one negating mirror
	R and M	same as above, i(M) = M R
p'bmm	t1 (t2)	pmm = {t1, t22, M}
	t2 and M	same as above, i(M) = M t2
	R and t1 (t2)	same as above, i(R) = R t1
	R, M and t1 (t2)	same as above, i(R) = R t1, i(M) = M t1
p'bgm	t1 and M	pmg = {t2, R, t12 M}; doubled cell
c'mm	t1 and t2	cmm = {t1 t2, R, M}; doubled cell
	M, t1 and t2	same as above, i(M) = M t2
	R, t1 and t2	same as above, i(R) = R t1
	R, M, t1 and t2	same as above, i(R) = R t1, i(M) = M t1

The Group pmg Ways to Visualize Wallpaper Functions Why there are 63 Types The Algebra of Wallpapers The Group pm