The pair group of a group 
is the group that acts on the 2-subsets of 
whose permutations
are induced by 
.
Pair groups can be calculated using PairGroup[g]
in the Wolfram Language package Combinatorica`
.
The cycle index for the pair group induced by 
is
![Z(S_p^((2)))=1/(p!)sum_((j))h_(j)product_(n=0)^(|_(p-1)/2_|)a_(2n+1)^(nj_(2n+1)+(2n+1)(j_(2n+1); 2))product_(n=1)^(|_p/2_|)[(a_na_(2n))^(n-1)]^(j_(2n))a_(2n)^(2n(j_(2n); 2))product_(q=1)^pproduct_(r=q+1)^pa_(LCM(q,r))^(j_qj_rGCD(q,r))](/images/equations/PairGroup/NumberedEquation1.svg) |
(1)
|
(Harary 1994, p. 185). Here, 
is the floor function,
is a binomial
coefficient, LCM is the least common multiple,
GCD is the greatest common divisor, the
sum 
is over all exponent vectors of the cycle index 
of the symmetric
group 
,
and 
is the coefficient of the term with
exponent vector 
in 
. The first few values of 
are
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
These can be given by PairGroup[SymmetricGroup[n], x] in the Wolfram Language package Combinatorica` .
