Theorem 3.1. The orthogonal subgroup OD
of a discrete subgroup D of Euclidean group preserves the lattice
RD.
(5) If two discrete subgroups of Euclidean group are isomorphic, then
their isomorphism preserves the type of isometries.
The function sending one group of isometries into another preserves
the type of transformations if it sends translations in translations, rotations
in rotations, reflections in reflections and glide reflections in glide
reflections. The proof follows: let D1 @y
D2, where D1, D2
Î DE2. As
isomorphism preserves the order, every rotation from D1
of an order higher than 2 goes into a rotation of the same order from D2.
Translations have no finite order, so that the image of a translation
from D1 must be either a translation or a glide reflection.
Let`s suppose that the translation t1Î
D1 goes to a glide reflection g2 Î
D2. Let t2 be the
translation from D2 which does not commute with g2
(every translation whose vector is not parallel with the glide reflection
axis will satisfy that condition). There exists x1 Î
D1, such that y(x1)
=
t2. Hence, x1
must be either a translation or a glide reflection. So x12
is a translation. As every two translations commute, we have
Because y is an isomorphism:
t22g2
= y(x12)y(t1)
= y(x12t1)
= y(t1x12)
= y(t1)y(x12)
= g2t22 |
|
This contradicts the fact that t2
and g2 don't commute.
A rotation of order 2 can have as it's y-image
a rotation of order 2 or a reflection. The same holds for reflections.
Let r1 be a rotation from D1 such that
y(r1) = s2,
where s2 is a reflection from D2. Let
t2 be a translation from D2,
such that s2t2
is a glide reflection. Now we have
y-1 (s2t2)
= y-1(s2)y-1(t2)
= r1t1, |
|
where t1 is the translation from
D1. Because the product of a rotation of order 2 and
the translation is a rotation of order 2, we will have here the isomorphism
which sends a glide reflection into a rotation, and that is the contradiction.
So we proved:
Theorem 3.2. The isomorphism of two subgroups of DE2
preserves the type of isometries.
From Theorem 3.2. follows that:
Corollary 3.1. If two subgroups of DE2
are isomorphic, their orthogonal groups are isomorphic as well.
Let D1, D2 Î
DE2, D1 @
fD2, and let d1
Î D1 be decomposed in
D1 in the product of two isometries. Then, from Theorem
3.2. follows that f(d1)
is D2-decomposable in two isometries.
Corollary 3.2. If two subgroups of DE2
are isomorphic, then their elements can be decomposed in the same types
of isometries.
Corollary 3.3. If two subgroups of DE2
are isomorphic, then the product of two elements from one group and the
product of their isomorphic images from the other group belong to the same
type of isometries.