Wolfgang Herfort

On automorphisms of free pro-

Let F be a (topologically) finitely generated free pro-p-group, and fi an automorphism of F . If p

p

2 and the order of fi is 2, then there is some basis of F such that fi either fixes or inverts its elements. If p does not divide the order of fi, then the subgroup of F of all elements fixed by fi is (topologically) infinitely generated; however this is not always the case if p divides the order of fi . Let p be a fixed prime number, and let F be a free pro-p-group of finite rank. In this paper we study (continuous) automorphisms of F. The group of automorphisms Aut(F) of F is, in a natural way, a profinite group. In [9J, Lubotzky gives global properties of the group Aut(F). Our interest here is rather more local; we describe properties of certain types of automorphisms. The group Aut(F) contains a pro-p-subgroup of finite index; hence the automorphisms of order prime to p must have finite order. In p

-groups. I

& 2, we show that given an automorphism p of order 2 of a finitely generated pro-p-group G (in particular, a free one), there is a minimal set of generators of G such that p sends each of the generators in that set to itself or its inverse. As a consequence we can describe all the conjugacy classes of involutions of Aut(F): they correspond bijectively to those of GL(n,p), where rankF = n. In [4], Gersten proves that if a is an automorphism of an abstract free group of finite rank, then the elements of the group fixed by a form a subgroup of finite rank also. In contrast, in ?3 we show that for a free pro-p-group F of finite rank, the equivalent result need not hold; in fact we prove that if the order of fi E Aut(F) is not divisible by p, and fi is not the identity, then the subgroup of the elements of F fixed by fi is necessarily infinitely generated (i.e. such a subgroup contains no dense subgroup which is finitely generated as an abstract group). This result depends strongly on the fact that the order of the automorphism does not involve the prime p. In fact, in ?4 we Received by the editors November 17, 1988 and, in revised form, February 13, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 20E18; 20F28. i) 1990 American Mathematical Society 0002-9939/90

Frobenius subgroups of free products of prosolvable groups

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p-Extensions of free pro-p groups

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Finite extensions of free pro-P groups of rank at most two

In this paper we establish the existence of profinite Frobenius subgroups in a free prosolvable productA ∐B of two finite groupsA andB. In this way the classification of solvable subgroups of free profinite groups is completed.

Deviations from Exponential Decay in the Spontaneous Emission by a Two-Level Hydrogenic Atom

Abstract It is proved that a virtually free pro-p group G having a free pro-p subgroup of index p satisfies a pro-p version of the Dyer-Scott structure theorem (Comm. Alg. 3(3) (1975), 195–201). The pro-2 case had been settled by W. Herfort and P. Zalesskii in (manuscr. math. 93, 457–464 (1997)). A proof for (topologically) finitely generated G has been given by C. Scheiderer. A consequence of our result is that for any automorphism of order pn of a free pro-p group its fixed point group is a free factor. The main theorem generalizes Serres well known result, stating that any virtually free torsion free pro-p group is free pro-p (Topology 3, 413–420, (1965)).

Topological Frobenius groups

For a pro-p groupG, containing a free pro-p open normal subgroup of rank at most 2, a characterization as the fundamental group of a connected graph of cyclic groups of order at mostp, and an explicit list of all such groups with trivial center are given. It is shown that any automorphism of a free pro-p group of rank 2 of coprime finite order is induced by an automorphism of the Frattini factor groupF/F*. Finally, a complete list of automorphisms of finite order, up to conjugacy in Aut(F), is given.

Abelian subgroups of topological groups

In our previous paper1 we have shown for the first time that neglecting of the retardation effects in the spontaneous emission from a two-level atom (dipole approximation) leads to an asymptotic result which differs significantly from that obtained without this approximation. For this reason in the present paper no kind of dipole or rotating-wave approximation (RWA) will be made. This can be achieved by specializing ourselves to the case of Lyman-a transition in a two-level hydrogenic atom.

The BCH-Formula and Order Conditions for Splitting Methods

Abstract Nonstandard analytic methods are applied to obtain results generalizing structure theorems of locally finite Frobenius groups and profinite Frobenius groups to the class of topologically locally finite, totally disconnected [ IN ]-groups. Uniform and neat proofs are given of Thompsons fixed point theorem in profinite groups and lifting theorems for fixed points of automorphism groups.