We give a new sucient condition for all zeros of self-inversive polynomials to be on the unit circle, and nd the location of zeros. This generalizes some recent results of Lakatos [7], Schinzel [17], Lakatos and Losonczi [9], [10]. By this sucient condition the mentioned results can be treated in a unied way.

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Let G be a finite p-group having a characteristic cyclic series (c.c.s.) and let Φ be its Frattini subgroup. It is shown that the automorphism group of G is either a p-group or is the semidirect product of a normal p-Sylow subgroup of G by an elementary abelian group of exponent p − 1 and of order (p − 1) r , where 1 ≤ r ≤ s and s = |G/Φ|. It is also shown that G has a c.c.s. containing Φ.such that each L i+1 /L i is cyclic. We consider finite p-groups, having cyclic characteristic series. If G is a finite p-group and it has a c.c.s. (1) then it has a characteristic composition series