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 n􀀀gon 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

Cp􀀀1 = 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.