Tarmizi, Rawdah Adawiyah
(2018)
Maximal Irredundant Coverings Of
Some Finite Groups.
PhD thesis, 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.
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