VW - CVM 1.1

# The Groups p3, p31m, p3m1, p6, and p6m

These five groups are intimately related, so we handle them all in a single table.

Generators:
 p3 = {t1, R3} p31m = {t1, R3, Mx} p3m1 = {t1, R3, My} p6 = {t1, R6} p6m = {t1, R6, Mx}
Relations:
t2 = R3 t1 R3-1,
R33 = e, R3-1 t1 R3 = t1-1 t2-1
R62 = R3, R6 Mx = My
Cell diagram:

The possibilities for homomorphisms from these groups is smaller than one might guess from the list of generators, because t1 and R3 can never be negating isometries of any pattern. To see this note that R cannot go to -1 because its order is odd, and that t1 cannot go to -1, because then t2 and the product t2 t1 would also be taken to -1, which would give a contradiction.

We group all the homomorphisms in one table.

 type P(k) = -1 wallpaper type of kernel group, with remarks p3 none E = G = p3 p31m none E = G = p31m p31m' Mx E = p31m, G = p3 = {t1, R3} p3m1 none E = G = p3m1 p3m' My E = p3m1, G = p3 p6 none E = G = p6 p6' R6 E = p6, G = p3 = {t1, R62} p6m none E = G = p6m p6m'm' Mx E = p6m, G = p6 p6'm'm R6 E = p6m, G = p31m = {t1, R62, Mx}; My negative p6'mm' R6 and Mx E = p6m, G = p3m1 = {t1, R62, R6 Mx}; My positive

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