An alternating group is a group of even permutations on a set of length 
,
denoted 
or Alt(
)
(Scott 1987, p. 267). Alternating groups are therefore permutation
groups.
The 
th
alternating group is represented in the Wolfram
Language as AlternatingGroup[n].
An alternating group is a normal subgroup of the permutation group, and has group
order 
,
the first few values of which for 
, 3, ... are 1, 3, 12, 60, 360, 2520, ... (OEIS A001710).
The alternating group 
is 
-transitive.
Amazingly, the pure rotational subgroup 
of the icosahedral group 
is isomorphic to 
. The full icosahedral group 
is isomorphic to the group
direct product 
,
where 
is the cyclic group on two elements.
Alternating groups 
with 
are simple groups (Scott 1987, p. 295), i.e.,
their only normal subgroups are the trivial subgroup and the entire group 
.
The number of conjugacy classes in the alternating groups 
for 
,
3, ... are 1, 3, 4, 5, 7, 9, ... (OEIS A000702).
is the only nontrivial proper normal subgroup of 
.
The multiplication table for 
is illustrated above.
The cycle index (in variables 
, ..., 
) for the alternating group 
is given by
![Z(A_p)=1/(p!)sum_((j))(p![1+(-1)^(j_2+j_4+...)])/(product_(k=1)^(p)k^(j_k)j_k!)a_1^(j_1)a_2^(j_2)...a_p^(j_p),](/images/equations/AlternatingGroup/NumberedEquation1.svg) |
(1)
|
(Harary 1994, p. 184), where the sum runs over the set of solution vectors 
to
 |
(2)
|
The cycle indices for the first few 
are
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
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(7)
|
