On X-Decomposable Finite Groups for X= {1,2,5}

Let [Formula: see text] be a finite group and [Formula: see text]. Then [Formula: see text] is called [Formula: see text]-decomposable in [Formula: see text] if it is a union of [Formula: see text] distinct [Formula: see text]-conjugacy classes, and denote [Formula: see text] by [Formula: see text]. A group [Formula: see text] is called [Formula: see text]-decomposable with [Formula: see text]. This paper proves that there are only five (isomorphism classes) [Formula: see text]-decomposable finite groups which are non-perfect.