An automorphism ,of the group G is said to be n-unipotent if [g, n , ] = 1 for all g 2 G. In this paper we obtain some results related to nilpotency of groups of n-unipotent AUTOMORPHISMS of solvable groups. We also show that, assuming the truth of a conjecture about the representation theory of solvable groups raised by P. Neumann, it is possible to produce, for a suitable prime p, an example of a f. g. solvable group possessing a group of p-unipotent AUTOMORPHISMS which is isomorphic to an in, nite Burnside group. Conversely we show that, if there exists a f. g. solvable group G with a non nilpotent p-group H of n-AUTOMORPHISMS, then there is such a counterexample where n is a prime power and H has , nite exponent.

We introduce the notions of commuting automorphism and commutator polygroups. The basic question that can be arose about the set of all commuting AUTOMORPHISMS is that for the assumed polygroup $ ( P, \cdot )$, under what conditions the set of all commuting automorphism $ \boldsymbol {A} ( P) $ is a subgroup of $ Aut ( P)$. In this paper basically the answer to this question is investigated for the class of polygroups.

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Let W be a non-empty subset of a free group. The automorphism a of a group G is said to be a marginal automorphism (with respect to W), if for all xÎG, x-1 a (x)ÎW*(G), where W*(G) is the marginal subgroup of G.In this paper, we study the concept of marginal AUTOMORPHISMS of a given group and we obtain a necessary and sufficient condition for a purely non-abelian finite p-group G, such that the set of all marginal AUTOMORPHISMS of G forms an elementary abelian p-group.

Let G be a group and AutΦ, (G) denote the group of all AUTOMORPHISMS of G centralizing G=Φ, (G) elementwise. In this paper, we give a necessary and sufficient condition on a , finite p-group G for the group AutΦ, (G) to be elementary abelian.

Let G be a  nite p-group. In this work we give the necessary and sufficient conditions on G such that each absolute central automorphism of G  xes the center element-wise. Also we classify all groups of the orders p3 and p4, whose absolute central AUTOMORPHISMS  x the center element-wise.