Abstract
Let [Formula: see text] be a finite [Formula: see text]-group and let [Formula: see text] be the set of all central automorphisms of [Formula: see text] For any group [Formula: see text], the center of the inner automorphism group, [Formula: see text], is always contained in [Formula: see text] In this paper, we study finite [Formula: see text]-groups [Formula: see text] for which [Formula: see text] is of minimal possible, that is [Formula: see text] We characterize the groups in some special cases, including [Formula: see text]-groups [Formula: see text] with [Formula: see text], [Formula: see text]-groups with an abelian maximal subgroup, metacyclic [Formula: see text]-groups with [Formula: see text], [Formula: see text]-groups of order [Formula: see text] and exponent [Formula: see text] and Camina [Formula: see text]-groups.
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