<s>
In	O
mathematics	O
,	O
an	O
alternating	B-Algorithm
group	I-Algorithm
is	O
the	O
group	O
of	O
even	O
permutations	B-Algorithm
of	O
a	O
finite	O
set	O
.	O
</s>
<s>
For	O
,	O
the	O
group	O
An	O
is	O
the	O
commutator	O
subgroup	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
Sn	O
with	O
index	O
2	O
and	O
has	O
therefore	O
n	O
!	O
/2	O
elements	O
.	O
</s>
<s>
It	O
is	O
the	O
kernel	O
of	O
the	O
signature	O
group	O
homomorphism	O
explained	O
under	O
symmetric	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
The	O
group	O
A4	O
has	O
the	O
Klein	O
four-group	O
V	O
as	O
a	O
proper	O
normal	O
subgroup	O
,	O
namely	O
the	O
identity	O
and	O
the	O
double	O
transpositions	B-Algorithm
,	O
that	O
is	O
the	O
kernel	O
of	O
the	O
surjection	B-Algorithm
of	O
A4	O
onto	O
.	O
</s>
<s>
As	O
in	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
,	O
any	O
two	O
elements	O
of	O
An	O
that	O
are	O
conjugate	O
by	O
an	O
element	O
of	O
An	O
must	O
have	O
the	O
same	O
cycle	O
shape	O
.	O
</s>
<s>
The	O
two	O
permutations	B-Algorithm
(	O
123	O
)	O
and	O
(	O
132	O
)	O
are	O
not	O
conjugates	O
in	O
A3	O
,	O
although	O
they	O
have	O
the	O
same	O
cycle	O
shape	O
,	O
and	O
are	O
therefore	O
conjugate	O
in	O
S3	O
.	O
</s>
<s>
The	O
permutation	B-Algorithm
(	O
123	O
)	O
(	O
45678	O
)	O
is	O
not	O
conjugate	O
to	O
its	O
inverse	O
(	O
132	O
)	O
(	O
48765	O
)	O
in	O
A8	O
,	O
although	O
the	O
two	O
permutations	B-Algorithm
have	O
the	O
same	O
cycle	O
shape	O
,	O
so	O
they	O
are	O
conjugate	O
in	O
S8	O
.	O
</s>
<s>
See	O
Symmetric	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
As	O
finite	O
symmetric	B-Algorithm
groups	I-Algorithm
are	O
the	O
groups	O
of	O
all	O
permutations	B-Algorithm
of	O
a	O
set	O
with	O
finite	O
elements	O
,	O
and	O
the	O
alternating	B-Algorithm
groups	I-Algorithm
are	O
groups	O
of	O
even	O
permutations	B-Algorithm
,	O
alternating	B-Algorithm
groups	I-Algorithm
are	O
subgroups	O
of	O
finite	O
symmetric	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
For	O
n	O
≥	O
3	O
,	O
An	O
is	O
generated	O
by	O
3-cycles	B-Algorithm
,	O
since	O
3-cycles	B-Algorithm
can	O
be	O
obtained	O
by	O
combining	O
pairs	O
of	O
transpositions	B-Algorithm
.	O
</s>
<s>
For	O
,	O
except	O
for	O
,	O
the	O
automorphism	B-Algorithm
group	I-Algorithm
of	O
An	O
is	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
Sn	O
,	O
with	O
inner	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
An	O
and	O
outer	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
Z2	O
;	O
the	O
outer	B-Algorithm
automorphism	I-Algorithm
comes	O
from	O
conjugation	O
by	O
an	O
odd	O
permutation	B-Algorithm
.	O
</s>
<s>
For	O
and	O
2	O
,	O
the	O
automorphism	B-Algorithm
group	I-Algorithm
is	O
trivial	O
.	O
</s>
<s>
For	O
the	O
automorphism	B-Algorithm
group	I-Algorithm
is	O
Z2	O
,	O
with	O
trivial	O
inner	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
and	O
outer	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
Z2	O
.	O
</s>
<s>
The	O
outer	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
of	O
A6	O
is	O
the	O
Klein	O
four-group	O
,	O
and	O
is	O
related	O
to	O
the	O
outer	B-Algorithm
automorphism	I-Algorithm
of	O
S6	O
.	O
</s>
<s>
The	O
extra	O
outer	B-Algorithm
automorphism	I-Algorithm
in	O
A6	O
swaps	O
the	O
3-cycles	B-Algorithm
(like(123 )	O
)	O
with	O
elements	O
of	O
shape	O
32	O
(	O
like	O
)	O
.	O
</s>
<s>
There	O
are	O
some	O
exceptional	B-Algorithm
isomorphisms	I-Algorithm
between	O
some	O
of	O
the	O
small	O
alternating	B-Algorithm
groups	I-Algorithm
and	O
small	O
groups	O
of	O
Lie	O
type	O
,	O
particularly	O
projective	O
special	O
linear	O
groups	O
.	O
</s>
<s>
The	O
nontrivial	O
outer	B-Algorithm
automorphism	I-Algorithm
in	O
interchanges	O
these	O
two	O
classes	O
and	O
the	O
corresponding	O
icosahedra	O
.	O
</s>
<s>
It	O
can	O
be	O
proved	O
that	O
the	O
15	O
puzzle	O
,	O
a	O
famous	O
example	O
of	O
the	O
sliding	O
puzzle	O
,	O
can	O
be	O
represented	O
by	O
the	O
alternating	B-Algorithm
group	I-Algorithm
A15	O
,	O
because	O
the	O
combinations	O
of	O
the	O
15	O
puzzle	O
can	O
be	O
generated	O
by	O
3-cycles	B-Algorithm
.	O
</s>
<s>
The	O
group	O
homology	O
of	O
the	O
alternating	B-Algorithm
groups	I-Algorithm
exhibits	O
stabilization	O
,	O
as	O
in	O
stable	O
homotopy	O
theory	O
:	O
for	O
sufficiently	O
large	O
n	O
,	O
it	O
is	O
constant	O
.	O
</s>
<s>
Note	O
that	O
the	O
homology	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
exhibits	O
similar	O
stabilization	O
,	O
but	O
without	O
the	O
low-dimensional	O
exceptions	O
(	O
additional	O
homology	O
elements	O
)	O
.	O
</s>
<s>
An	O
is	O
generated	O
by	O
3-cycles	B-Algorithm
–	O
so	O
the	O
only	O
non-trivial	O
abelianization	O
maps	O
are	O
since	O
order-3	O
elements	O
must	O
map	O
to	O
order-3	O
elements	O
–	O
and	O
for	O
all	O
3-cycles	B-Algorithm
are	O
conjugate	O
,	O
so	O
they	O
must	O
map	O
to	O
the	O
same	O
element	O
in	O
the	O
abelianization	O
,	O
since	O
conjugation	O
is	O
trivial	O
in	O
abelian	O
groups	O
.	O
</s>
<s>
For	O
A3	O
and	O
A4	O
one	O
can	O
compute	O
the	O
abelianization	O
directly	O
,	O
noting	O
that	O
the	O
3-cycles	B-Algorithm
form	O
two	O
conjugacy	O
classes	O
(	O
rather	O
than	O
all	O
being	O
conjugate	O
)	O
and	O
there	O
are	O
non-trivial	O
maps	O
(	O
in	O
fact	O
an	O
isomorphism	O
)	O
and	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
5	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>
