## Derivation of Cycle Index Formulas of Semidirect Product Groups

##### Abstract

The concept of the cycle index formulas of a permutation group was discovered in
the year 1937. Since then cycle index formulas of several groups have been studied
by di erent scholars. For instance the cycle index of the dihedral group Dn acting on
the set of vertices of a regular ngon is known and has been applied in enumeration
of di erent mathematical structures. In this study the relationship between the cycle
index formula of a semidirect product group and the cycle index formulas of the two
subgroups which the group is a semidirect product of was established. In particular
the cycle index formula of the dihedral group Dn of order 2n is expressed in terms
of the cycle index formula of a cyclic group of order two C2 and the cycle index
formula of the cyclic group of order n, Cn; the cycle index formula of the symmetric
group Sn is expressed in terms of the cycle index formula of the alternating group
An and the cycle index formula of a group generated by a cycle of length two, h(ab)i.
The cycle index formula of an a ne(p) group has been derived by considering the
di erent cycle types of elements of the group and expressed in terms of the cycle
index formula of Cp = fx + b; where b 2 Zpg and the cycle index formula of
Cp1 = fax; where 0 6= a 2 Zpg. We further extend this to a ne(q) where q is a
power of a prime p and to the a ne square(p) and a ne square(q) groups. Finally,
the cycle index formula of a Frobenius group is expressed in terms of the cycle index
formula of the Frobenius complement H and the cycle index formula of the Frobenius
kernel M. The cycle index formulas which are known such as that of the dihedral
group and the symmetric group were used and the groups whose cycle index formulas
are not known such as the a ne(p), a ne square(p); a ne(q) and a ne square(q)
group were rst derived as part of the research. It was noted that for semidirect
groups which are Frobenius such as the dihedral group Dn with an odd value of n,
the a ne groups and the a ne square groups, we can fully express the cycle index
of the group in terms of the cycle index formulas of the subgroups which the group
is a semidirect of. However, for semidirect product groups which are not Frobenius
such as the dihedral group Dn with an even value of n and the symmetric group Sn,
the cycle index formula of the group cannot be expressed fully in terms of the cycle
index formulas of the subgroups the group is a semidirect product of.