<s>
In	O
mathematics	O
,	O
the	O
generalized	B-Algorithm
symmetric	I-Algorithm
group	I-Algorithm
is	O
the	O
wreath	O
product	O
of	O
the	O
cyclic	O
group	O
of	O
order	O
m	O
and	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
of	O
order	O
n	O
.	O
</s>
<s>
For	O
the	O
generalized	B-Algorithm
symmetric	I-Algorithm
group	I-Algorithm
is	O
exactly	O
the	O
ordinary	O
symmetric	B-Algorithm
group	I-Algorithm
:	O
</s>
<s>
For	O
one	O
can	O
consider	O
the	O
cyclic	O
group	O
of	O
order	O
2	O
as	O
positives	O
and	O
negatives	O
(	O
)	O
and	O
identify	O
the	O
generalized	B-Algorithm
symmetric	I-Algorithm
group	I-Algorithm
with	O
the	O
signed	B-Algorithm
symmetric	I-Algorithm
group	I-Algorithm
.	O
</s>
<s>
There	O
is	O
a	O
natural	O
representation	O
of	O
elements	O
of	O
as	O
generalized	B-Algorithm
permutation	I-Algorithm
matrices	I-Algorithm
,	O
where	O
the	O
nonzero	O
entries	O
are	O
m-th	O
roots	O
of	O
unity	O
:	O
</s>
<s>
As	O
with	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
,	O
the	O
representations	O
can	O
be	O
constructed	O
in	O
terms	O
of	O
Specht	O
modules	O
;	O
see	O
.	O
</s>
<s>
The	O
first	O
group	O
homology	O
group	O
(	O
concretely	O
,	O
the	O
abelianization	O
)	O
is	O
(	O
for	O
m	O
odd	O
this	O
is	O
isomorphic	O
to	O
)	O
:	O
the	O
factors	O
(	O
which	O
are	O
all	O
conjugate	O
,	O
hence	O
must	O
map	O
identically	O
in	O
an	O
abelian	O
group	O
,	O
since	O
conjugation	O
is	O
trivial	O
in	O
an	O
abelian	O
group	O
)	O
can	O
be	O
mapped	O
to	O
(	O
concretely	O
,	O
by	O
taking	O
the	O
product	O
of	O
all	O
the	O
values	O
)	O
,	O
while	O
the	O
sign	O
map	O
on	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
yields	O
the	O
These	O
are	O
independent	O
,	O
and	O
generate	O
the	O
group	O
,	O
hence	O
are	O
the	O
abelianization	O
.	O
</s>
<s>
Note	O
that	O
it	O
depends	O
on	O
n	O
and	O
the	O
parity	O
of	O
m	O
:	O
and	O
which	O
are	O
the	O
Schur	O
multipliers	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
and	O
signed	B-Algorithm
symmetric	I-Algorithm
group	I-Algorithm
.	O
</s>
