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

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

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

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.

Table: Eleven homomorphisms from groups related to p3
type P(k) = -1 wallpaper type of kernel group, with remarks
p3 [Image] [Image] none E = G = p3
p31m [Image] none E = G = p31m
p31m' [Image] Mx E = p31m, G = p3 = {t1, R3}
p3m1 [Image] none E = G = p3m1
p3m' [Image] My E = p3m1, G = p3
p6 [Image] none E = G = p6
p6' [Image] R6 E = p6, G = p3 = {t1, R62}
p6m [Image] none E = G = p6m
p6m'm' [Image] Mx E = p6m, G = p6
p6'm'm [Image] R6 E = p6m, G = p31m = {t1, R62, Mx}; My negative
p6'mm' [Image] R6 and Mx E = p6m, G = p3m1 = {t1, R62, R6 Mx}; My positive

[Next] Ways to Visualize Wallpaper Functions
[Up] Why there are 63 Types
[UpUp] The Algebra of Wallpapers
[Prev] The Group p4m

Communications in Visual Mathematics, vol 1, no 1, August 1998.
Copyright © 1998, The Mathematical Association of America. All rights reserved.
Created: 08 Jul 1998 --- Last modified: 18 Aug 1998 23:59:59
Comments to: CVM@maa.org