S-PERMUTABILITY, SEMIPERMUTABILITY, c-NORMALITY AND c-PERMUTABILITY OF SUBGROUPS IN FINITE GROUPS
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Date
2014-08
Authors
Al-Sharo, Doa a Mustafa
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Abstract
All groups and subgroups considered in this thesis are nite. In this thesis,
some new theories and related proofs on S-permutable (also referred to as Sylow per-
mutable), semipermutable, c-normal and c-permutable subgroups of a nite group G
are discussed. More than one topic are focused here, some of which are, the relationship
between classes of S-permutable embedded subgroups and PST-groups, the semiper-
mutable subgroups, and the condition when c-normal and c-permutable subgroups are
transitive. New ndings reveal that PST-groups, the class of groups in which S-
permutability is a transitive relation, coincides with the class of nite solvable groups
in which every p-subgroup is S-permutably embedded. Some facts were developed to
prove this result, in particular, a simpli ed proof is given to prove that if a subgroup
is S-permutably embedded in the group then it should be S-permutably embedded in
every subgroup that contains it. A dual result is also established for a group called
the PSTc-group where "c" denotes the special condition on the nilpotent residual.
Research on semipermutability of subgroups and their transitivity relationships have
been extensively explored in previous years. When semipermutability in a group G is
a transitive relation, G is called a BT-group. When all subnormal subgroups of G are
semipermutable, G is called an SP-group. It has been shown that all BT-groups are
SP-groups. However, an SP-group is not necessarily a BT-group as shown in the the-
sis. For the condition in which c-normality is a transitive relation, a new class of group
called CT-groups is introduced and some related properties and theorems to this group
are proved. Some new results related to CPT-groups, which are groups in which c-
permutability is a transitive relation, are also given. This thesis also contains the proof
that there exists a c-permutable subgroup which is not c-normal. New characteriza-
tions of the following groups and subgroups are de ned and proved in this thesis which
are: the nite solvable PST-groups, S-permutably embedded subgroups, BT-groups,
semipermutable subgroups, seminormal subgroups, nite solvable CT-groups, mini-
mal non-CT-groups, minimal non-CPT-groups and groups with all its subnormal sub-
groups c-normal (called SCN groups), and groups with all its subnormal subgroups c-
permutable (called SCp-groups).
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Keywords
S-PERMUTABILITY , SEMIPERMUTABILITY , c-NORMALITY , c-PERMUTABILITY