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Supplements in finite groups.

McGough, Robert John 1971 , 'Supplements in finite groups.', Unspecified thesis, University of Tasmania.

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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
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]).

Item Type: Thesis - Unspecified
Authors/Creators:McGough, Robert John
Keywords: Finite groups
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Copyright 1971 the Author - The University is continuing to endeavour to trace the copyright
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Thesis (M.Sc.) - University of Tasmania, 1971

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