<s>
In	O
the	O
mathematical	O
area	O
of	O
group	O
theory	O
,	O
the	O
covering	B-Algorithm
groups	I-Algorithm
of	I-Algorithm
the	I-Algorithm
alternating	I-Algorithm
and	I-Algorithm
symmetric	I-Algorithm
groups	I-Algorithm
are	O
groups	O
that	O
are	O
used	O
to	O
understand	O
the	O
projective	O
representations	O
of	O
the	O
alternating	B-Algorithm
and	O
symmetric	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
The	O
covering	O
groups	O
were	O
classified	O
in	O
:	O
for	O
,	O
the	O
covering	O
groups	O
are	O
2-fold	O
covers	O
except	O
for	O
the	O
alternating	B-Algorithm
groups	I-Algorithm
of	O
degree	O
6	O
and	O
7	O
where	O
the	O
covers	O
are	O
6-fold	O
.	O
</s>
<s>
For	O
example	O
the	O
binary	O
icosahedral	O
group	O
covers	O
the	O
icosahedral	O
group	O
,	O
an	O
alternating	B-Algorithm
group	I-Algorithm
of	O
degree	O
5	O
,	O
and	O
the	O
binary	O
tetrahedral	O
group	O
covers	O
the	O
tetrahedral	O
group	O
,	O
an	O
alternating	B-Algorithm
group	I-Algorithm
of	O
degree	O
4	O
.	O
</s>
<s>
The	O
Schur	O
covers	O
of	O
the	O
symmetric	O
and	O
alternating	B-Algorithm
groups	I-Algorithm
were	O
classified	O
in	O
.	O
</s>
<s>
The	O
alternating	B-Algorithm
group	I-Algorithm
of	O
degree	O
n	O
has	O
one	O
isomorphism	O
class	O
of	O
Schur	O
cover	O
,	O
which	O
has	O
order	O
n	O
!	O
</s>
<s>
These	O
relations	O
can	O
be	O
used	O
to	O
describe	O
two	O
non-isomorphic	O
covers	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
Sometimes	O
all	O
of	O
the	O
relations	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
are	O
expressed	O
as	O
(	O
titj	O
)	O
mij	O
=	O
1	O
,	O
where	O
mij	O
are	O
non-negative	O
integers	O
,	O
namely	O
mii	O
=	O
1	O
,	O
mi	O
,	O
i+1	O
=	O
3	O
,	O
and	O
mij	O
=	O
2	O
,	O
for	O
1	O
≤	O
i	O
i+2	O
≤	O
j	O
≤	O
n−1	O
.	O
</s>
<s>
The	O
projective	O
representations	O
of	O
alternating	B-Algorithm
and	O
symmetric	B-Algorithm
groups	I-Algorithm
are	O
the	O
subject	O
of	O
the	O
book	O
.	O
</s>
<s>
The	O
Schur	O
multipliers	O
of	O
the	O
alternating	B-Algorithm
groups	I-Algorithm
An	O
(	O
in	O
the	O
case	O
where	O
n	O
is	O
at	O
least	O
4	O
)	O
are	O
the	O
cyclic	O
groups	O
of	O
order	O
2	O
,	O
except	O
in	O
the	O
case	O
where	O
n	O
is	O
either	O
6	O
or	O
7	O
,	O
in	O
which	O
case	O
there	O
is	O
also	O
a	O
triple	O
cover	O
.	O
</s>
<s>
For	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
,	O
the	O
Schur	O
multiplier	O
vanishes	O
for	O
n	O
≤	O
3	O
,	O
and	O
is	O
the	O
cyclic	O
group	O
of	O
order	O
2	O
for	O
n	O
≥	O
4	O
:	O
</s>
<s>
Explicitly	O
,	O
Sn	O
acts	O
on	O
the	O
n-dimensional	O
space	O
Rn	O
by	O
permuting	O
coordinates	O
(	O
in	O
matrices	O
,	O
as	O
permutation	B-Algorithm
matrices	I-Algorithm
)	O
.	O
</s>
<s>
The	O
special	O
orthogonal	O
group	O
has	O
a	O
2-fold	O
cover	O
by	O
the	O
spin	O
group	O
and	O
restricting	O
this	O
cover	O
to	O
and	O
taking	O
the	O
preimage	O
yields	O
a	O
2-fold	O
cover	O
A	O
similar	O
construction	O
with	O
a	O
pin	O
group	O
yields	O
the	O
2-fold	O
cover	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
:	O
As	O
there	O
are	O
two	O
pin	O
groups	O
,	O
there	O
are	O
two	O
distinct	O
2-fold	O
covers	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
,	O
2⋅Sn±	O
,	O
also	O
called	O
and	O
.	O
</s>
<s>
For	O
low	O
dimensions	O
there	O
are	O
exceptional	B-Algorithm
isomorphisms	I-Algorithm
with	O
the	O
map	O
from	O
a	O
special	O
linear	O
group	O
over	O
a	O
finite	O
field	O
to	O
the	O
projective	O
special	O
linear	O
group	O
.	O
</s>
<s>
For	O
n	O
=	O
3	O
,	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
is	O
SL(2,2 )	O
≅	O
PSL(2,2 )	O
and	O
is	O
its	O
own	O
Schur	O
cover	O
.	O
</s>
<s>
For	O
n	O
=	O
4	O
,	O
the	O
Schur	O
cover	O
of	O
the	O
alternating	B-Algorithm
group	I-Algorithm
is	O
given	O
by	O
SL(2,3 )	O
→	O
PSL(2,3 )	O
≅	O
A4	O
,	O
which	O
can	O
also	O
be	O
thought	O
of	O
as	O
the	O
binary	O
tetrahedral	O
group	O
covering	O
the	O
tetrahedral	O
group	O
.	O
</s>
<s>
For	O
n	O
=	O
5	O
,	O
the	O
Schur	O
cover	O
of	O
the	O
alternating	B-Algorithm
group	I-Algorithm
is	O
given	O
by	O
SL(2,5 )	O
→	O
PSL(2,5 )	O
≅	O
A5	O
,	O
which	O
can	O
also	O
be	O
thought	O
of	O
as	O
the	O
binary	O
icosahedral	O
group	O
covering	O
the	O
icosahedral	O
group	O
.	O
</s>
<s>
For	O
n	O
=	O
6	O
,	O
the	O
double	O
cover	O
of	O
the	O
alternating	B-Algorithm
group	I-Algorithm
is	O
given	O
by	O
SL(2,9 )	O
→	O
PSL(2,9 )	O
≅	O
A6	O
.	O
</s>
<s>
The	O
Schur	O
covers	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
S6	O
itself	O
have	O
no	O
faithful	O
representations	O
as	O
a	O
subgroup	O
of	O
GL(d,9 )	O
for	O
d≤3	O
.	O
</s>
<s>
For	O
n	O
=	O
8	O
,	O
the	O
alternating	B-Algorithm
group	I-Algorithm
A8	O
is	O
isomorphic	O
to	O
SL(4,2 )	O
=	O
PSL(4,2 )	O
,	O
and	O
so	O
SL(4,2 )	O
→	O
PSL(4,2 )	O
,	O
which	O
is	O
1-to-1	O
,	O
not	O
2-to-1	O
,	O
is	O
not	O
a	O
Schur	O
cover	O
.	O
</s>
