### Abstract:

Let G be a group and p a prime number. G
is said to be a Yp-group if whenever K is a p-subgroup of G then
every subgroup of K is an S-permutable subgroup in NG(K). The
group G is a soluble PST-group if and only if G is a Yp-group for
all primes p.
One of our purposes here is to define a number of local proper-
ties related to Yp which lead to several new characterizations of
soluble PST-groups. Another purpose is to define several embed-
ding subgroup properties which yield some new classes of soluble
PST-groups. Such properties include weakly S-permutable subgroup,
weakly semipermutable subgroup, and weakly seminormal subgroup.