Maximal Irredundant Coverings Of Some Finite Groups
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Date
2018-08
Authors
Rawdah Adawiyah Tarmizi
Journal Title
Journal ISSN
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Publisher
Universiti Sains Malaysia
Abstract
The aim of this research is to contribute further results on the coverings of some
finite groups. Only non-cyclic groups are considered in the study of group coverings.
Since no group can be covered by only two of its proper subgroups, a covering should
consist of at least 3 of its proper subgroups. If a covering contains n (proper) subgroups,
then the set of these subgroups is called an n-covering. The covering of a
group G is called minimal if it consists of the least number of proper subgroups among
all coverings for the group; i.e. if the minimal covering consists of m proper subgroups
then the notation used is s(G) = m. A covering of a group is called irredundant if no
proper subset of the covering also covers the group. Obviously, every minimal covering
is irredundant but the converse is not true in general. If the members of the covering
are all maximal normal subgroups of a group G, then the covering is called a maximal
covering. Let D be the intersection of all members in the covering. Then the covering
is said to have core-free intersection if the core of D is the trivial subgroup. A maximal
irredundant n-covering with core-free intersection is known as a Cn-covering and a
group with this type of covering is known as a Cn-group. This study focuses only on the
minimal covering of the symmetric group S9 and the dihedral group Dn for odd n 3;
on the characterization of p-groups having a Cn-covering for n 2 f10;11;12g; and the
characterization of nilpotent groups having a Cn-covering for n 2 f9;10;11;12g. In
this thesis, a lower bound and an upper bound for s(S9) is established. (However,
later it was found that the exact value for s(S9) = 256 has already been discovered in 2016.) For the dihedral groups Dn where n is odd and n 3, results were presented in
two classifications, i.e. the prime n and the odd composite n. For the p-groups, it was
found that the only p-groups with Cn-coverings for n 2 f10;11;12g are those isomorphic
to some elementary abelian groups of certain orders and the results established the
concrete proof of the groups. It was also found that some p-groups have all three possible
types of coverings and some others have two of the three types of coverings. For
the nilpotent groups, it was found that for n 2 f10;11;12g, the nilpotent groups having
Cn-coverings are exactly the p-groups obtained earlier; no other nilpotent groups
were found to have Cn-coverings for n 2 f10;11;12g. The nilpotent groups having a
C9-covering are also isomorphic to some elementary abelian groups of certain orders.
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Keywords
Mathematics