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The Group pmm

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

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 [Image] none pmm, same as the extended group
pm'm' [Image] 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 [Image] 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 [Image] 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


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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: Sep 30, 2003 9:26:33 AM
Comments to: CVM@maa.org