Supplements in finite groups.

A well-known problem in the theory of groups is that of finding conditions under which a given subgroup of a finite group has a normal complement. Since a group may be written as a semi-direct product of a subgroup and its normal complement, the structure of a group may be determined from the structure of these two smaller groups. Hence solutions of the above problem may be useful in determining the structure of certain groups. Important early work on normal complements was done by Burnside ([2], p.203) and Frobenius ([2], p.292). More recently Kochendorffer [5], Sah [10], Suzuki [12] and Zappa [15] have worked on this problem. The theorem of Suzuki has as corollaries the theorems of Burnside, Frobenius and Zappa. Kochendorffer [6] introduced the concept of a coefficient group belonging to a set of coset representatives of a given subgroup. Using this concept he has obtained results on the existence of a normal supplement to a given subgroup which are similar to those obtained on the existence of a normal complement. These results have since been extended by Prohaska [9]. In this thesis we introduce the concept of D-closure and use it to obtain several results on the existence of a normal supplement to a subgroup of a finite group. In Chapter 3 we show that D-closure is a weaker condition than that of Kockendorffer and that his and Prohaska's theorems still hold under the weaker condition. We then note that 1)-closure is a generalization of the concept of c-closure introduced by Sah [10] and obtain direct generalizations of results due to him and Suzuki. The major part of this thesis is contained in Chapter 3; Chapter 2 contains only introductory notation and definitions and a discussion of some of the properties of c-closure. Since completing the work on this thesis, section 3.2 has been published in the Bulletin of the Australian Mathematical Society under the title \A Note on a Theorem of Sah\" (see [7])."