<s>
The	O
two-dimensional	O
point	O
groups	O
are	O
important	O
as	O
a	O
basis	O
for	O
the	O
axial	O
three-dimensional	B-Algorithm
point	I-Algorithm
groups	I-Algorithm
,	O
with	O
the	O
addition	O
of	O
reflections	O
in	O
the	O
axial	O
coordinate	O
.	O
</s>
<s>
Abstract	O
group	O
Dihn	O
,	O
the	O
dihedral	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
In	O
the	O
infinite	O
limit	O
,	O
these	O
groups	O
become	O
the	O
one-dimensional	O
line	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
Generated	O
by	O
element	O
Cn	O
and	O
reflection	B-Algorithm
.	O
</s>
<s>
All	O
of	O
the	O
cyclic	O
groups	O
are	O
abelian	O
or	O
commutative	O
,	O
but	O
only	O
two	O
of	O
the	O
dihedral	B-Algorithm
groups	I-Algorithm
are	O
:	O
D1	O
~	O
Z2	O
and	O
D2	O
~	O
Z2Z2	O
.	O
</s>
<s>
The	O
n	O
in	O
the	O
H-M	O
symbol	O
denotes	O
n-fold	O
rotations	O
,	O
while	O
the	O
m	O
denotes	O
reflection	B-Algorithm
or	O
mirror	B-Algorithm
planes	I-Algorithm
.	O
</s>
<s>
These	O
groups	O
are	O
readily	O
constructed	O
with	O
two-dimensional	O
orthogonal	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
The	O
continuous	O
dihedral	B-Algorithm
group	I-Algorithm
O(2 )	O
or	O
D∞	O
and	O
its	O
subgroups	O
with	O
reflections	O
have	O
elements	O
that	O
include	O
not	O
only	O
rotation	O
matrices	O
,	O
but	O
also	O
reflection	B-Algorithm
matrices	O
:	O
</s>
<s>
For	O
every	O
subgroup	O
of	O
SO(2 )	O
there	O
is	O
a	O
corresponding	O
uncountable	O
class	O
of	O
subgroups	O
of	O
O(2 )	O
that	O
are	O
mutually	O
isomorphic	O
as	O
abstract	O
group	O
:	O
each	O
of	O
the	O
subgroups	O
in	O
one	O
class	O
is	O
generated	O
by	O
the	O
first-mentioned	O
subgroup	O
and	O
a	O
single	O
reflection	B-Algorithm
in	O
a	O
line	O
through	O
the	O
origin	O
.	O
</s>
<s>
These	O
are	O
the	O
(	O
generalized	O
)	O
dihedral	B-Algorithm
groups	I-Algorithm
,	O
including	O
the	O
finite	O
ones	O
Dn	O
(	O
n	O
≥	O
1	O
)	O
of	O
abstract	O
group	O
type	O
Dihn	O
.	O
</s>
<s>
The	O
symmetry	O
group	O
of	O
a	O
square	O
belongs	O
to	O
the	O
family	O
of	O
dihedral	B-Algorithm
groups	I-Algorithm
,	O
Dn	O
(	O
abstract	O
group	O
type	O
Dihn	O
)	O
,	O
including	O
as	O
many	O
reflections	O
as	O
rotations	O
.	O
</s>
<s>
The	O
infinite	O
rotational	O
symmetry	O
of	O
the	O
circle	O
implies	O
reflection	B-Algorithm
symmetry	O
as	O
well	O
,	O
but	O
formally	O
the	O
circle	O
group	O
S1	O
is	O
distinct	O
from	O
Dih(S1 )	O
because	O
the	O
latter	O
explicitly	O
includes	O
the	O
reflections	O
.	O
</s>
<s>
There	O
is	O
a	O
"	O
natural	O
"	O
surjective	B-Algorithm
group	O
homomorphism	O
p	O
:	O
E(2 )	O
→	O
E(2 )	O
/	O
T	O
,	O
sending	O
each	O
element	O
g	O
of	O
E(2 )	O
to	O
the	O
coset	O
of	O
T	O
to	O
which	O
g	O
belongs	O
,	O
that	O
is	O
:	O
p	O
(	O
g	O
)	O
=	O
gT	O
,	O
sometimes	O
called	O
the	O
canonical	O
projection	O
of	O
E(2 )	O
onto	O
E(2 )	O
/	O
T	O
or	O
O(2 )	O
.	O
</s>
<s>
Thus	O
these	O
10	O
groups	O
give	O
rise	O
to	O
17	O
wallpaper	O
groups	O
,	O
and	O
the	O
four	O
groups	O
with	O
n	O
=	O
1	O
and	O
2	O
,	O
give	O
also	O
rise	O
to	O
7	O
frieze	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
the	O
"	O
projections	O
"	O
onto	O
E(2 )	O
/	O
T	O
or	O
O(2 )	O
)	O
are	O
all	O
equal	O
to	O
the	O
corresponding	O
Cn	O
;	O
also	O
two	O
frieze	B-Algorithm
groups	I-Algorithm
correspond	O
to	O
C1	O
and	O
C2	O
.	O
</s>
<s>
Also	O
five	O
frieze	B-Algorithm
groups	I-Algorithm
correspond	O
to	O
D1	O
and	O
D2	O
.	O
</s>
<s>
For	O
each	O
of	O
the	O
types	O
D1	O
,	O
D2	O
,	O
and	O
D4	O
the	O
distinction	O
between	O
the	O
3	O
,	O
4	O
,	O
and	O
2	O
wallpaper	O
groups	O
,	O
respectively	O
,	O
is	O
determined	O
by	O
the	O
translation	O
vector	O
associated	O
with	O
each	O
reflection	B-Algorithm
in	O
the	O
group	O
:	O
since	O
isometries	O
are	O
in	O
the	O
same	O
coset	O
regardless	O
of	O
translational	O
components	O
,	O
a	O
reflection	B-Algorithm
and	O
a	O
glide	B-Algorithm
reflection	I-Algorithm
with	O
the	O
same	O
mirror	O
are	O
in	O
the	O
same	O
coset	O
.	O
</s>
<s>
This	O
is	O
because	O
the	O
conjugate	O
of	O
the	O
translation	O
by	O
a	O
glide	B-Algorithm
reflection	I-Algorithm
is	O
the	O
same	O
as	O
by	O
the	O
corresponding	O
reflection	B-Algorithm
:	O
the	O
translation	O
vector	O
is	O
reflected	O
.	O
</s>
