<s>
In	O
mathematics	O
,	O
a	O
dihedral	B-Algorithm
group	I-Algorithm
is	O
the	O
group	O
of	O
symmetries	O
of	O
a	O
regular	O
polygon	O
,	O
which	O
includes	O
rotations	O
and	O
reflections	B-Algorithm
.	O
</s>
<s>
Dihedral	B-Algorithm
groups	I-Algorithm
are	O
among	O
the	O
simplest	O
examples	O
of	O
finite	O
groups	O
,	O
and	O
they	O
play	O
an	O
important	O
role	O
in	O
group	O
theory	O
,	O
geometry	O
,	O
and	O
chemistry	O
.	O
</s>
<s>
The	O
notation	O
for	O
the	O
dihedral	B-Algorithm
group	I-Algorithm
differs	O
in	O
geometry	O
and	O
abstract	O
algebra	O
.	O
</s>
<s>
In	O
abstract	O
algebra	O
,	O
refers	O
to	O
this	O
same	O
dihedral	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
A	O
regular	O
polygon	O
with	O
sides	O
has	O
different	O
symmetries	O
:	O
rotational	O
symmetries	O
and	O
reflection	B-Algorithm
symmetries	I-Algorithm
.	O
</s>
<s>
The	O
associated	O
rotations	O
and	O
reflections	B-Algorithm
make	O
up	O
the	O
dihedral	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
If	O
is	O
even	O
,	O
there	O
are	O
axes	B-Algorithm
of	I-Algorithm
symmetry	I-Algorithm
connecting	O
the	O
midpoints	O
of	O
opposite	O
sides	O
and	O
axes	B-Algorithm
of	I-Algorithm
symmetry	I-Algorithm
connecting	O
opposite	O
vertices	O
.	O
</s>
<s>
In	O
either	O
case	O
,	O
there	O
are	O
axes	B-Algorithm
of	I-Algorithm
symmetry	I-Algorithm
and	O
elements	O
in	O
the	O
symmetry	O
group	O
.	O
</s>
<s>
The	O
first	O
row	O
shows	O
the	O
effect	O
of	O
the	O
eight	O
rotations	O
,	O
and	O
the	O
second	O
row	O
shows	O
the	O
effect	O
of	O
the	O
eight	O
reflections	B-Algorithm
,	O
in	O
each	O
case	O
acting	O
on	O
the	O
stop	O
sign	O
with	O
the	O
orientation	O
as	O
shown	O
at	O
the	O
top	O
left	O
.	O
</s>
<s>
r0	O
denotes	O
the	O
identity	O
;	O
r1	O
and	O
r2	O
denote	O
counterclockwise	O
rotations	O
by	O
120°	O
and	O
240°	O
respectively	O
,	O
and	O
s0	O
,	O
s1	O
and	O
s2	O
denote	O
reflections	B-Algorithm
across	O
the	O
three	O
lines	O
shown	O
in	O
the	O
adjacent	O
picture	O
.	O
</s>
<s>
For	O
example	O
,	O
,	O
because	O
the	O
reflection	B-Algorithm
s1	O
followed	O
by	O
the	O
reflection	B-Algorithm
s2	O
results	O
in	O
a	O
rotation	O
of	O
120°	O
.	O
</s>
<s>
If	O
we	O
center	O
the	O
regular	O
polygon	O
at	O
the	O
origin	O
,	O
then	O
elements	O
of	O
the	O
dihedral	B-Algorithm
group	I-Algorithm
act	O
as	O
linear	B-Architecture
transformations	I-Architecture
of	O
the	O
plane	O
.	O
</s>
<s>
This	O
lets	O
us	O
represent	O
elements	O
of	O
Dn	O
as	O
matrices	B-Architecture
,	O
with	O
composition	O
being	O
matrix	O
multiplication	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
elements	O
of	O
the	O
group	O
D4	O
can	O
be	O
represented	O
by	O
the	O
following	O
eight	O
matrices	B-Architecture
:	O
</s>
<s>
In	O
general	O
,	O
the	O
matrices	B-Architecture
for	O
elements	O
of	O
Dn	O
have	O
the	O
following	O
form	O
:	O
</s>
<s>
rk	O
is	O
a	O
rotation	B-Algorithm
matrix	I-Algorithm
,	O
expressing	O
a	O
counterclockwise	O
rotation	O
through	O
an	O
angle	O
of	O
.	O
</s>
<s>
sk	O
is	O
a	O
reflection	B-Algorithm
across	O
a	O
line	O
that	O
makes	O
an	O
angle	O
of	O
with	O
the	O
x-axis	O
.	O
</s>
<s>
and	O
are	O
the	O
only	O
abelian	O
dihedral	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
is	O
a	O
subgroup	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
for	O
.	O
</s>
<s>
The	O
inner	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
of	O
is	O
trivial	O
,	O
whereas	O
for	O
other	O
even	O
values	O
of	O
,	O
this	O
is	O
.	O
</s>
<s>
The	O
cycle	O
graphs	O
of	O
dihedral	B-Algorithm
groups	I-Algorithm
consist	O
of	O
an	O
n-element	O
cycle	O
and	O
n	O
2-element	O
cycles	O
.	O
</s>
<s>
The	O
dark	O
vertex	O
in	O
the	O
cycle	O
graphs	O
below	O
of	O
various	O
dihedral	B-Algorithm
groups	I-Algorithm
represents	O
the	O
identity	O
element	O
,	O
and	O
the	O
other	O
vertices	O
are	O
the	O
other	O
elements	O
of	O
the	O
group	O
.	O
</s>
<s>
These	O
groups	O
form	O
one	O
of	O
the	O
two	O
series	O
of	O
discrete	O
point	B-Algorithm
groups	I-Algorithm
in	I-Algorithm
two	I-Algorithm
dimensions	I-Algorithm
.	O
</s>
<s>
consists	O
of	O
rotations	O
of	O
multiples	O
of	O
about	O
the	O
origin	O
,	O
and	O
reflections	B-Algorithm
across	O
lines	O
through	O
the	O
origin	O
,	O
making	O
angles	O
of	O
multiples	O
of	O
with	O
each	O
other	O
.	O
</s>
<s>
(	O
Compare	O
coordinate	B-Algorithm
rotations	I-Algorithm
and	I-Algorithm
reflections	I-Algorithm
.	O
)	O
</s>
<s>
The	O
dihedral	B-Algorithm
group	I-Algorithm
D2	I-Algorithm
is	O
generated	O
by	O
the	O
rotation	O
r	O
of	O
180	O
degrees	O
,	O
and	O
the	O
reflection	B-Algorithm
s	B-Algorithm
across	O
the	O
x-axis	O
.	O
</s>
<s>
The	O
elements	O
of	O
D2	O
can	O
then	O
be	O
represented	O
as	O
{e,r,s,rs},	O
where	O
e	O
is	O
the	O
identity	O
or	O
null	O
transformation	O
and	O
rs	O
is	O
the	O
reflection	B-Algorithm
across	O
the	O
y-axis	O
.	O
</s>
<s>
For	O
n	O
>	O
2	O
the	O
operations	O
of	O
rotation	O
and	O
reflection	B-Algorithm
in	O
general	O
do	O
not	O
commute	O
and	O
Dn	O
is	O
not	O
abelian	O
;	O
for	O
example	O
,	O
in	O
D4	O
,	O
a	O
rotation	O
of	O
90	O
degrees	O
followed	O
by	O
a	O
reflection	B-Algorithm
yields	O
a	O
different	O
result	O
from	O
a	O
reflection	B-Algorithm
followed	O
by	O
a	O
rotation	O
of	O
90	O
degrees	O
.	O
</s>
<s>
The	O
first	O
listed	O
elements	O
are	O
rotations	O
and	O
the	O
remaining	O
elements	O
are	O
axis-reflections	O
(	O
all	O
of	O
which	O
have	O
order2	O
)	O
.	O
</s>
<s>
The	O
product	O
of	O
two	O
rotations	O
or	O
two	O
reflections	B-Algorithm
is	O
a	O
rotation	O
;	O
the	O
product	O
of	O
a	O
rotation	O
and	O
a	O
reflection	B-Algorithm
is	O
a	O
reflection	B-Algorithm
.	O
</s>
<s>
the	O
group	O
of	O
rotations	O
(	O
about	O
the	O
origin	O
)	O
and	O
reflections	B-Algorithm
(	O
across	O
axes	O
through	O
the	O
origin	O
)	O
of	O
the	O
plane	O
.	O
</s>
<s>
Therefore	O
,	O
it	O
is	O
also	O
called	O
a	O
dihedron	O
(	O
Greek	O
:	O
solid	O
with	O
two	O
faces	O
)	O
,	O
which	O
explains	O
the	O
name	O
dihedral	B-Algorithm
group	I-Algorithm
(	O
in	O
analogy	O
to	O
tetrahedral	O
,	O
octahedral	O
and	O
icosahedral	O
group	O
,	O
referring	O
to	O
the	O
proper	O
symmetry	O
groups	O
of	O
a	O
regular	O
tetrahedron	O
,	O
octahedron	O
,	O
and	O
icosahedron	O
respectively	O
)	O
.	O
</s>
<s>
The	O
properties	O
of	O
the	O
dihedral	B-Algorithm
groups	I-Algorithm
with	O
depend	O
on	O
whether	O
is	O
even	O
or	O
odd	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
center	O
of	O
consists	O
only	O
of	O
the	O
identity	O
if	O
n	O
is	O
odd	O
,	O
but	O
if	O
n	O
is	O
even	O
the	O
center	O
has	O
two	O
elements	O
,	O
namely	O
the	O
identity	O
and	O
the	O
element	O
rn/2	O
(	O
with	O
Dn	O
as	O
a	O
subgroup	O
of	O
O(2 )	O
,	O
this	O
is	O
inversion	B-Algorithm
;	O
since	O
it	O
is	O
scalar	O
multiplication	O
by	O
1	O
,	O
it	O
is	O
clear	O
that	O
it	O
commutes	O
with	O
any	O
linear	B-Architecture
transformation	I-Architecture
)	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
2D	O
isometries	O
,	O
this	O
corresponds	O
to	O
adding	O
inversion	B-Algorithm
,	O
giving	O
rotations	O
and	O
mirrors	O
in	O
between	O
the	O
existing	O
ones	O
.	O
</s>
<s>
The	O
dihedral	B-Algorithm
group	I-Algorithm
of	O
order	O
8	O
(	O
D4	O
)	O
is	O
the	O
smallest	O
example	O
of	O
a	O
group	O
that	O
is	O
not	O
a	O
T-group	O
.	O
</s>
<s>
Any	O
of	O
its	O
two	O
Klein	O
four-group	O
subgroups	O
(	O
which	O
are	O
normal	O
in	O
D4	O
)	O
has	O
as	O
normal	O
subgroup	O
order-2	O
subgroups	O
generated	O
by	O
a	O
reflection	B-Algorithm
(	O
flip	B-Algorithm
)	O
in	O
D4	O
,	O
but	O
these	O
subgroups	O
are	O
not	O
normal	O
in	O
D4	O
.	O
</s>
<s>
All	O
the	O
reflections	B-Algorithm
are	O
conjugate	O
to	O
each	O
other	O
whenever	O
n	O
is	O
odd	O
,	O
but	O
they	O
fall	O
into	O
two	O
conjugacy	O
classes	O
if	O
n	O
is	O
even	O
.	O
</s>
<s>
Algebraically	O
,	O
this	O
is	O
an	O
instance	O
of	O
the	O
conjugate	O
Sylow	O
theorem	O
(	O
for	O
n	O
odd	O
)	O
:	O
for	O
n	O
odd	O
,	O
each	O
reflection	B-Algorithm
,	O
together	O
with	O
the	O
identity	O
,	O
form	O
a	O
subgroup	O
of	O
order	O
2	O
,	O
which	O
is	O
a	O
Sylow	O
2-subgroup	O
(	O
is	O
the	O
maximum	O
power	O
of	O
2	O
dividing	O
)	O
,	O
while	O
for	O
n	O
even	O
,	O
these	O
order	O
2	O
subgroups	O
are	O
not	O
Sylow	O
subgroups	O
because	O
4	O
(	O
a	O
higher	O
power	O
of	O
2	O
)	O
divides	O
the	O
order	O
of	O
the	O
group	O
.	O
</s>
<s>
For	O
n	O
even	O
there	O
is	O
instead	O
an	O
outer	B-Algorithm
automorphism	I-Algorithm
interchanging	O
the	O
two	O
types	O
of	O
reflections	B-Algorithm
(	O
properly	O
,	O
a	O
class	O
of	O
outer	B-Algorithm
automorphisms	I-Algorithm
,	O
which	O
are	O
all	O
conjugate	O
by	O
an	O
inner	B-Algorithm
automorphism	I-Algorithm
)	O
.	O
</s>
<s>
The	O
automorphism	B-Algorithm
group	I-Algorithm
of	O
is	O
isomorphic	O
to	O
the	O
holomorph	B-Algorithm
of	O
/n	O
,	O
i.e.	O
,	O
to	O
and	O
has	O
order	O
nϕ(n )	O
,	O
where	O
ϕ	O
is	O
Euler	O
's	O
totient	O
function	O
,	O
the	O
number	O
of	O
k	O
in	O
coprime	O
to	O
n	O
.	O
</s>
<s>
It	O
can	O
be	O
understood	O
in	O
terms	O
of	O
the	O
generators	O
of	O
a	O
reflection	B-Algorithm
and	O
an	O
elementary	O
rotation	O
(	O
rotation	O
by	O
k( 	O
2π/n	O
)	O
,	O
for	O
k	O
coprime	O
to	O
n	O
)	O
;	O
which	O
automorphisms	O
are	O
inner	O
and	O
outer	O
depends	O
on	O
the	O
parity	O
of	O
n	O
.	O
</s>
<s>
For	O
n	O
odd	O
,	O
the	O
dihedral	B-Algorithm
group	I-Algorithm
is	O
centerless	O
,	O
so	O
any	O
element	O
defines	O
a	O
non-trivial	O
inner	B-Algorithm
automorphism	I-Algorithm
;	O
for	O
n	O
even	O
,	O
the	O
rotation	O
by	O
180°	O
(	O
reflection	B-Algorithm
through	O
the	O
origin	O
)	O
is	O
the	O
non-trivial	O
element	O
of	O
the	O
center	O
.	O
</s>
<s>
Thus	O
for	O
n	O
odd	O
,	O
the	O
inner	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
has	O
order	O
2n	O
,	O
and	O
for	O
n	O
even	O
(	O
other	O
than	O
)	O
the	O
inner	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
has	O
order	O
n	O
.	O
</s>
<s>
For	O
n	O
odd	O
,	O
all	O
reflections	B-Algorithm
are	O
conjugate	O
;	O
for	O
n	O
even	O
,	O
they	O
fall	O
into	O
two	O
classes	O
(	O
those	O
through	O
two	O
vertices	O
and	O
those	O
through	O
two	O
faces	O
)	O
,	O
related	O
by	O
an	O
outer	B-Algorithm
automorphism	I-Algorithm
,	O
which	O
can	O
be	O
represented	O
by	O
rotation	O
by	O
π/n	O
(	O
half	O
the	O
minimal	O
rotation	O
)	O
.	O
</s>
<s>
The	O
rotations	O
are	O
a	O
normal	O
subgroup	O
;	O
conjugation	O
by	O
a	O
reflection	B-Algorithm
changes	O
the	O
sign	O
(	O
direction	O
)	O
of	O
the	O
rotation	O
,	O
but	O
otherwise	O
leaves	O
them	O
unchanged	O
.	O
</s>
<s>
has	O
18	O
inner	B-Algorithm
automorphisms	I-Algorithm
.	O
</s>
<s>
The	O
18	O
inner	B-Algorithm
automorphisms	I-Algorithm
provide	O
rotation	O
of	O
the	O
mirrors	O
by	O
multiples	O
of	O
20°	O
,	O
and	O
reflections	B-Algorithm
.	O
</s>
<s>
As	O
abstract	O
group	O
there	O
are	O
in	O
addition	O
to	O
these	O
,	O
36	O
outer	B-Algorithm
automorphisms	I-Algorithm
;	O
e.g.	O
,	O
multiplying	O
angles	O
of	O
rotation	O
by	O
2	O
.	O
</s>
<s>
has	O
10	O
inner	B-Algorithm
automorphisms	I-Algorithm
.	O
</s>
<s>
The	O
10	O
inner	B-Algorithm
automorphisms	I-Algorithm
provide	O
rotation	O
of	O
the	O
mirrors	O
by	O
multiples	O
of	O
36°	O
,	O
and	O
reflections	B-Algorithm
.	O
</s>
<s>
As	O
isometry	O
group	O
there	O
are	O
10	O
more	O
automorphisms	O
;	O
they	O
are	O
conjugates	O
by	O
isometries	O
outside	O
the	O
group	O
,	O
rotating	O
the	O
mirrors	O
18°	O
with	O
respect	O
to	O
the	O
inner	B-Algorithm
automorphisms	I-Algorithm
.	O
</s>
<s>
As	O
abstract	O
group	O
there	O
are	O
in	O
addition	O
to	O
these	O
10	O
inner	O
and	O
10	O
outer	B-Algorithm
automorphisms	I-Algorithm
,	O
20	O
more	O
outer	B-Algorithm
automorphisms	I-Algorithm
;	O
e.g.	O
,	O
multiplying	O
rotations	O
by	O
3	O
.	O
</s>
<s>
Compare	O
the	O
values	O
6	O
and	O
4	O
for	O
Euler	O
's	O
totient	O
function	O
,	O
the	O
multiplicative	O
group	O
of	O
integers	O
modulo	O
n	O
for	O
n	O
=	O
9	O
and	O
10	O
,	O
respectively	O
.	O
</s>
<s>
The	O
only	O
values	O
of	O
n	O
for	O
which	O
φ(n )	O
=	O
2	O
are	O
3	O
,	O
4	O
,	O
and	O
6	O
,	O
and	O
consequently	O
,	O
there	O
are	O
only	O
three	O
dihedral	B-Algorithm
groups	I-Algorithm
that	O
are	O
isomorphic	O
to	O
their	O
own	O
automorphism	B-Algorithm
groups	I-Algorithm
,	O
namely	O
(	O
order	O
6	O
)	O
,	O
(	O
order	O
8	O
)	O
,	O
and	O
(	O
order	O
12	O
)	O
.	O
</s>
<s>
The	O
inner	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
of	O
is	O
isomorphic	O
to	O
:	O
</s>
<s>
There	O
are	O
several	O
important	O
generalizations	O
of	O
the	O
dihedral	B-Algorithm
groups	I-Algorithm
:	O
</s>
<s>
The	O
infinite	O
dihedral	B-Algorithm
group	I-Algorithm
is	O
an	O
infinite	O
group	O
with	O
algebraic	O
structure	O
similar	O
to	O
the	O
finite	O
dihedral	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
The	O
orthogonal	O
group	O
O(2 )	O
,	O
i.e.	O
,	O
the	O
symmetry	O
group	O
of	O
the	O
circle	O
,	O
also	O
has	O
similar	O
properties	O
to	O
the	O
dihedral	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
The	O
family	O
of	O
generalized	O
dihedral	B-Algorithm
groups	I-Algorithm
includes	O
both	O
of	O
the	O
examples	O
above	O
,	O
as	O
well	O
as	O
many	O
other	O
groups	O
.	O
</s>
<s>
The	O
quasidihedral	O
groups	O
are	O
family	O
of	O
finite	O
groups	O
with	O
similar	O
properties	O
to	O
the	O
dihedral	B-Algorithm
groups	I-Algorithm
.	O
</s>
