<s>
In	O
mathematics	O
,	O
especially	O
in	O
the	O
area	O
of	O
algebra	O
known	O
as	O
group	O
theory	O
,	O
the	O
holomorph	B-Algorithm
of	O
a	O
group	O
is	O
a	O
group	O
that	O
simultaneously	O
contains	O
(	O
copies	O
of	O
)	O
the	O
group	O
and	O
its	O
automorphism	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
The	O
holomorph	B-Algorithm
provides	O
interesting	O
examples	O
of	O
groups	O
,	O
and	O
allows	O
one	O
to	O
treat	O
group	O
elements	O
and	O
group	O
automorphisms	O
in	O
a	O
uniform	O
context	O
.	O
</s>
<s>
In	O
group	O
theory	O
,	O
for	O
a	O
group	O
,	O
the	O
holomorph	B-Algorithm
of	O
denoted	O
can	O
be	O
described	O
as	O
a	O
semidirect	O
product	O
or	O
as	O
a	O
permutation	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
For	O
the	O
holomorph	B-Algorithm
,	O
and	O
is	O
the	O
identity	O
map	O
,	O
as	O
such	O
we	O
suppress	O
writing	O
explicitly	O
in	O
the	O
multiplication	O
given	O
in	O
[	O
Eq	O
.	O
</s>
<s>
and	O
this	O
group	O
is	O
not	O
abelian	O
,	O
as	O
,	O
so	O
that	O
is	O
a	O
non-abelian	O
group	O
of	O
order	O
6	O
,	O
which	O
,	O
by	O
basic	O
group	O
theory	O
,	O
must	O
be	O
isomorphic	O
to	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
A	O
group	O
G	O
acts	O
naturally	O
on	O
itself	O
by	O
left	O
and	O
right	O
multiplication	O
,	O
each	O
giving	O
rise	O
to	O
a	O
homomorphism	O
from	O
G	O
into	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
on	O
the	O
underlying	O
set	O
of	O
G	O
.	O
One	O
homomorphism	O
is	O
defined	O
as	O
λ	O
:	O
G	O
→	O
Sym(G )	O
,	O
(	O
h	O
)	O
=	O
g·h	O
.	O
</s>
<s>
That	O
is	O
,	O
g	O
is	O
mapped	O
to	O
the	O
permutation	B-Algorithm
obtained	O
by	O
left-multiplying	O
each	O
element	O
of	O
G	O
by	O
g	O
.	O
Similarly	O
,	O
a	O
second	O
homomorphism	O
ρ	O
:	O
G	O
→	O
Sym(G )	O
is	O
defined	O
by	O
(	O
h	O
)	O
=	O
h·g−1	O
,	O
where	O
the	O
inverse	O
ensures	O
that	O
(	O
k	O
)	O
=	O
((k )	O
)	O
.	O
</s>
<s>
These	O
homomorphisms	O
are	O
called	O
the	O
left	O
and	O
right	O
regular	O
representations	O
of	O
G	O
.	O
Each	O
homomorphism	O
is	O
injective	O
,	O
a	O
fact	O
referred	O
to	O
as	O
Cayley	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
.	O
</s>
<s>
The	O
image	O
of	O
λ	O
is	O
a	O
subgroup	O
of	O
Sym(G )	O
isomorphic	O
to	O
G	O
,	O
and	O
its	O
normalizer	O
in	O
Sym(G )	O
is	O
defined	O
to	O
be	O
the	O
holomorph	B-Algorithm
N	O
of	O
G	O
.	O
</s>
<s>
If	O
an	O
element	O
n	O
of	O
the	O
holomorph	B-Algorithm
fixes	O
the	O
identity	O
of	O
G	O
,	O
then	O
for	O
1	O
in	O
G	O
,	O
(	O
n·	O
)	O
(	O
1	O
)	O
=	O
( ·n	O
)	O
(	O
1	O
)	O
,	O
but	O
the	O
left	O
hand	O
side	O
is	O
n(g )	O
,	O
and	O
the	O
right	O
side	O
is	O
h	O
.	O
In	O
other	O
words	O
,	O
if	O
n	O
in	O
N	O
fixes	O
the	O
identity	O
of	O
G	O
,	O
then	O
for	O
every	O
g	O
in	O
G	O
,	O
n·	O
=	O
λ(n(g )	O
)	O
·n	O
.	O
</s>
<s>
Since	O
is	O
transitive	O
,	O
the	O
subgroup	O
generated	O
by	O
and	O
the	O
point	O
stabilizer	O
A	O
is	O
all	O
of	O
N	O
,	O
which	O
shows	O
the	O
holomorph	B-Algorithm
as	O
a	O
permutation	B-Algorithm
group	I-Algorithm
is	O
isomorphic	O
to	O
the	O
holomorph	B-Algorithm
as	O
semidirect	O
product	O
.	O
</s>
<s>
since	O
λ(g )	O
ρ(g )	O
(	O
h	O
)	O
=	O
ghg−1	O
(	O
is	O
the	O
group	O
of	O
inner	B-Algorithm
automorphisms	I-Algorithm
of	O
G	O
.	O
)	O
</s>
