<s>
In	O
abstract	O
algebra	O
,	O
a	O
branch	O
of	O
mathematics	O
,	O
the	O
algebraic	O
structure	O
group	B-Algorithm
with	I-Algorithm
operators	I-Algorithm
or	O
Ω-group	O
can	O
be	O
viewed	O
as	O
a	O
group	O
with	O
a	O
set	O
Ω	O
that	O
operates	O
on	O
the	O
elements	O
of	O
the	O
group	O
in	O
a	O
special	O
way	O
.	O
</s>
<s>
A	O
group	B-Algorithm
with	I-Algorithm
operators	I-Algorithm
can	O
be	O
defined	O
as	O
a	O
group	O
together	O
with	O
an	O
action	O
of	O
a	O
set	O
on	O
:	O
</s>
<s>
In	O
category	O
theory	O
,	O
a	O
group	B-Algorithm
with	I-Algorithm
operators	I-Algorithm
can	O
be	O
defined	O
as	O
an	O
object	O
of	O
a	O
functor	B-Language
category	O
GrpM	O
where	O
M	O
is	O
a	O
monoid	O
(	O
i.e.	O
</s>
<s>
A	O
morphism	O
in	O
this	O
category	O
is	O
a	O
natural	O
transformation	O
between	O
two	O
functors	B-Language
(	O
i.e.	O
,	O
two	O
groups	O
with	O
operators	O
sharing	O
same	O
operator	O
domain	O
M	O
)	O
.	O
</s>
<s>
Given	O
a	O
module	O
M	O
over	O
a	O
ring	O
R	O
,	O
R	O
acts	O
by	O
scalar	O
multiplication	O
on	O
the	O
underlying	O
abelian	O
group	O
of	O
M	O
,	O
so	O
(	O
M	O
,	O
R	O
)	O
is	O
a	O
group	B-Algorithm
with	I-Algorithm
operators	I-Algorithm
.	O
</s>
<s>
As	O
a	O
special	O
case	O
of	O
the	O
above	O
,	O
every	O
vector	O
space	O
over	O
a	O
field	O
k	O
is	O
a	O
group	B-Algorithm
with	I-Algorithm
operators	I-Algorithm
(	O
V	O
,	O
k	O
)	O
.	O
</s>
<s>
The	O
requirement	O
that	O
a	O
group	O
have	O
a	O
composition	O
series	O
is	O
analogous	O
to	O
that	O
of	O
compactness	O
in	O
topology	B-Architecture
,	O
and	O
can	O
sometimes	O
be	O
too	O
strong	O
a	O
requirement	O
.	O
</s>
