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ABSTRACT
Let be a finite group. Given a conjugacy class
of
, the covering number of
,
, is the minimal natural number
(if exists) for which
. The covering number of
is
is a non-trivial conjugacy class of
. It is known that the above covering numbers exist if
is nonabelian and simple. However, if
is non-perfect, these covering numbers do not exist. In this paper we extend the concept of covering number to any finite group. Then we study the family of groups having a conjugacy class
of involutions such that
, and the family of groups having a conjugacy class
for which
is a prime power and
.