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Right 2-Engel Elements and Commuting Automorphisms of Groups
Journal of Algebra 238, 479᎐484 Ž2001. doi:10.1006rjabr.2000.8655, available online at http:rrwww.idealibrary.com on
Right 2-Engel Elements and Commuting Automorphisms of Groups Marian Deaconescu1 Department of Mathematics and Computer Science, Kuwait Uni¨ ersity, P.O. Box 5969, Safat 13060, Kuwait
and Gary L. Walls Department of Mathematics, Uni¨ ersity of Southern Mississippi, Hattiesburg, Mississippi 39401 Communicated by Georgia Benkart Received December 13, 1999
It is shown that there is a close connection between the right 2-Engel elements of a group and the set of the so-called commuting automorphisms of the group. As a consequence, the following general theorem is proved: If G is a group and if R 2 Ž G . denotes the subgroup of right 2-Engel elements, then the factor group X R 2 Ž G . l CG Ž G .rZ2 Ž G . is a group of exponent at most 2. 䊚 2001 Academic Press Key Words: commuting automorphisms; central automorphisms; right 2-Engel elements
INTRODUCTION Let G be a group and consider the set R 2 Ž G . s g g G ¬ w g, x, x x s 1 for all x g G4 of right 2-Engel elements of G. It is well known, see Kappe w5x, that R 2 Ž G . is a characteristic subgroup of G. It is also known, see Heineken w3x, that the inverse of a right 2-Engel element is a left 3-Engel element. 1 While working at this paper, the first named author was supported by Research Grant SM 09r00 from Kuwait University Research Administration.
479 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
480
DEACONESCU AND WALLS
In this paper we are interested in the structure of the subgroup R 2 Ž G . for centerless groups G. Results from w1x and w2x imply that if G is a finite group with trivial center, then R 2 Ž G . s 1. One would expect that whenever G is a group with trivial center R 2 Ž G . is either trivial, or has a structure which is hard to control; indeed, the structure of R 2 Ž G . is known to be under control provided that some rather strong conditions are imposed on G, as, for example, the condition that G satisfies the maximum condition on subgroupsᎏsee, for example, Baer w1x or the discussion in w9, Sect. 12.3x. In a certain sense, centerless groups are as far from being nilpotent as one could get; recent results, see w7x, show that even in nilpotent groups there may exist 5-Engel elements none of whose powers is a left 5-Engel element. The following result comes then as a surprise; it asserts that all groups possess a certain canonic section which has a very special structure: THEOREM 1.1. Let G be a group. Then R 2 Ž G . l CG Ž GX .rZ2 Ž G . is a group of exponent at most 2. In particular, if ZŽ G . s 1, then R 2 Ž G . is a subgroup of G of exponent at most 2. The above theorem is a consequence of results related to the so-called commuting automorphisms of groups. An automorphism ␣ of a group G is called a commuting automorphism if g ␣ Ž g . s ␣ Ž g . g for all g g G. The set of all commuting automorphisms of G is denoted here by AŽ G . and it is clear that set AŽ G . contains the group Autc Ž G . of central automorphisms of G. The converse inclusion does not hold in general; in fact, see w2x, there exist finite nonabelian 2-groups G such that AŽ G . is not a subgroup of Aut Ž G .. Previous results suggest, however, the following: Conjecture. If G is a group and if Autc Ž G . s id G 4 , then AŽ G . s id G 4 . This conjecture was shown to be true in the following particular cases: G is a simple nonabelian group ŽI. N. Herstein w4x., or G has no nontrivial abelian normal subgroups ŽT. J. Laffey w6x., or G s GX and ZŽ G . s 1 ŽM. Pettet w8x., or G is finite ŽDeaconescu et al. w2x.. The second main result of this paper shows that the above conjecture is false: THEOREM 1.2. There exist infinite groups G such that ZŽ G . s 1 and AŽ G . / id G 4 . More information about the groups whose existence is ensured by Theorem 1.2 is contained in THEOREM 1.3.
Let G be a group such that ZŽ G . s 1 and AŽ G . / id G 4 .
Ži. AŽ G . is a subgroup of Aut Ž G . and expŽ AŽ G .. s 2. Žii. 1 - R 2 Ž G . F CG Ž GX . and expŽ R 2 Ž G .. s 2.
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Žiii. If ␣ g AŽ G ., then ␣ fixes G 2 R 2 Ž G ., where G 2 is the subgroup of G generated by the set of squares of elements in G. The above result has an immediate consequence which shows that if a group G has no nontrivial central automorphisms, then the structure of the set AŽ G . is very simple: COROLLARY 1.4. If G is a group such that Autc Ž G . s id G 4 , then AŽ G . is a subgroup of Aut Ž G . of exponent at most 2. It is well known that if G is a group with trivial center, then the center of Aut Ž G . is trivial too. Our last result shows that the same property holds for R 2 Ž G .: If G is a group such that R 2 Ž G . is tri¨ ial, then R 2 Ž Aut Ž G ..
THEOREM 1.5. is tri¨ ial.
PRELIMINARY LEMMAS The first lemma is due to T. J. Laffey w6x and is very useful in computations: LEMMA 2.1.
Let ␣ g AŽ G .. Then w ␣ Ž x ., y x s w x, ␣ Ž y .x for all x, y g G.
The next result bridges together the right 2-Engel elements and the commuting automorphisms of a group: LEMMA 2.2. Ži. If ␣ g AŽ G . and if x g G, then xy1␣ Ž x . g R 2 Ž G .. Žii. Let g g G and denote by Tg the inner automorphism induced by g. Then Tg g AŽ G . if and only if g g R 2 Ž G .. In particular, InnŽ G . l AŽ G . is isomorphic to R 2 Ž G .rZŽ G .. Proof. The proofs of both assertions are elementary and can be found in w2x. The following properties of commuting automorphisms are essential for what follows: LEMMA 2.3. Ži. Žii.
Let ␣ g AŽ G ..
If x g GX , then xy1␣ Ž x . g ZŽ G .. GX : CG Ž ␣ . if and only if ␣ 2 g Autc Ž G ..
Proof. See w2x.
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DEACONESCU AND WALLS
The following result provides a criterion for AŽ G . to be trivial: LEMMA 2.4. Let G be a group with no nontri¨ ial central automorphisms. Then AŽ G . s id G 4 if and only if R 2 Ž G . s ZŽ G .. In particular, if G s GX , then R 2 Ž G . s ZŽ G .. Proof. If AŽ G . s id G 4 , then R 2 Ž G . s ZŽ G . by Lemma 2.2Žii.. If R 2 Ž G . s ZŽ G ., it follows from Lemma 2.2Ži. that AŽ G . : Autc Ž G ., whence AŽ G . s id G 4 . To prove the last assertion assume that G s GX . Then Autc Ž G . is trivial since central automorphisms of G fix GX elementwise. Lemma 2.3Ži. implies that AŽ G . is trivial. Thus R 2 Ž G . s ZŽ G . by the above paragraph. PROOFS OF THE MAIN RESULTS Proof of Theorem 1.2. We will construct an infinite group G with trivial center and such that R 2 Ž G . is nontrivial. Since ZŽ G . s 1, it follows at once that Autc Ž G . is trivial; the fact that R 2 Ž G . is nontrivial implies, via Lemma 2.4, that AŽ G . is not trivial. Let H denote the direct sum Žrestricted direct product. of copies of the cyclic group of order two, indexed over the set of positive integers. Let G denote the standard restricted wreath product of Z2 by H. That is, G is the semidirect product of the base group B Žwhich is the direct sum of < H < copies of Z2 . by H by using the regular representation of H to define the action of H on B. In this representation each nontrivial element of H acts as a formal product of disjoint 2-cycles which fixes no point in B. Also, B has a Z2-basis Žcorresponding to the elements of H . on which H acts transitively. It is known that any standard wreath product of an abelian group by an infinite group is centerless, thus ZŽ G . is trivial. Let now x s Ž1, 0, 0, . . . . g B and let y s bh be an arbitrary element of G with b g B and h g H. Assuming that h contains the 2-cycle Ž1, i ., then x y s x h s Ž0, 0, . . . , 1, 0, 0, . . . . because 1 and 0 are interchanged and all other interchanges are between zeros. Then w x, y x s Ž1, 0, 0, 0, . . . , 1, 0, 0, . . . . and y commutes with w x, y x since b does and since h will interchange the two 1’s and a set of zeros. It follows that Ž1, 0, 0, . . . . is a right 2-Engel element, as are all tuples with only one nonzero component. But these elements generate B. Thus B is contained in R 2 Ž G . and since R 2 Ž G . is abelian it follows that B s R 2 Ž G .. Thus here R 2 Ž G . is an infinite group of exponent 2 and this completes the proof of Theorem 2.1. Remark. The above considerations show that all the elements with exactly two 1’s are in GX . These elements generate the subgroup of B of elements which have an even number of 1’s. That is clearly a normal
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subgroup of G with abelian quotient. So GX is this subgroup and therefore GX centralizes B s R 2 Ž G .. This last property of our example is not accidental; it will turn out that R 2 Ž G . centralizes GX in every group G with trivial center. Proof of Theorem 1.3. Since ZŽ G . s 1, it follows at once that Autc Ž G . is trivial. By hypothesis, AŽ G . is nontrivial, so by Lemma 2.4, R 2 Ž G . / 1. Fix a nontrivial automorphism ␣ g AŽ G .. Since ZŽ G . s 1, Lemma 2.3Ži. implies that ␣ fixes GX and by Lemma 2.3Žii. one concludes that ␣ 2 s id G . This shows that expŽ AŽ G .. s 2. For x g G, we have that xy1␣ Ž x . g R 2 Ž G . by Lemma 2.2Ži.. Now since Txy1 ␣ Ž x . g AŽ G . by Lemma 2.2Žii., one obtains that the order of this inner automorphism is at most 2. Hence Ž xy1␣ Ž x .. 2 s 1 and since x ␣ Ž x . s ␣ Ž x . x it follows that ␣ Ž x 2 . s x 2 . Thus the subgroup G 2 s ² g 2 ¬ g g G4: is fixed by ␣ . Take now r g R 2 Ž G ., so that Tr g AŽ G . by Lemma 2.2Žii. and Tr2 s id G . Then r 2 g ZŽ G . s 1 and we obtain that expŽ R 2 Ž G .. s 2. Moreover, Lemma 2.3Žii. implies that Tr fixes GX , whence r g CG Ž GX .. Hence R 2 Ž G . F CG Ž GX . and the proof of part Žii. of the theorem is complete. To finish the proof of part Ži., note that AŽ G . is closed with respect to taking inverses. Let ␣ ,  g AŽ G . and let x g G. Then, since xy1␣ Ž x ., xy1 Ž x . g R 2 Ž G . and since R 2 Ž G . is abelian, we derive, using Lemma 2.1, that 1 s w xy1␣ Ž x ., xy1 Ž x .x s w ␣ Ž x .,  Ž x .x s w x, ␣ Ž x .x, which shows that ␣ g AŽ G .. Thus AŽ G . is a subgroup of Aut Ž G ., which proves Ži.. To complete the proof of part Žiii., we only need to show that every ␣ g AŽ G . fixes R 2 Ž G .. Take r g R 2 Ž G ., so that the inner automorphism Tr g AŽ G .. Since by part Ži., AŽ G . is abelian, it follows that w Tr , ␣ x s id G . Since ZŽ G . s 1, a short calculation shows that ␣ Ž r . s r. Proof of Theorem 1.1. Let r g R 2 Ž G . l CG Ž GX .. Then the inner automorphism Tr is a commuting automorphism of G which fixes GX , so by Lemma 2.3Žii., Tr2 g Autc Ž G . l InnŽ G .. But then, since Autc Ž G . l InnŽ G . s ZŽ InnŽ G .. ( Z2 Ž G .rZŽ G ., it follows at once that r 2 g Z2 Ž G .. This proves the first assertion of Theorem 1.1. If ZŽ G . s 1, then Z2 Ž G . s 1 and by Theorem 1.3Žii., R 2 Ž G . F CG Ž GX .. Thus R 2 Ž G . ( R 2 Ž G . l CG Ž GX .rZŽ G . is a group of exponent at most 2 by the first part of the theorem. The proof is now complete. Proof of Corollary 1.4. If G has no nontrivial central automorphisms, then necessarily Z2 Ž G . s ZŽ G .. Set G s GrZŽ G . and observe that every ␣ g AŽ G . induces an automorphism ␣ g AŽ G .. But since G has trivial center, it follows from Theorem 1.3Ži. that ␣ 2 s id G . Now the function : Aut Ž G . ª Aut Ž G ., defined by Ž ␣ . s ␣ , is one-by-one since Ker s Autc Ž G . s id G 4 . Thus AŽ G . is embedded into AŽ G . and the result follows from Theorem 1.3Ži..
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Proof of Theorem 1.5. We claim first that if G is a group with ZŽ G . s 1, then R 2 Ž Aut Ž G .. : AŽ G .. Indeed, let ␣ g R 2 Ž Aut Ž G ... Then w ␣ , Tx , Tx x s id G for every Tx g InnŽ G .. Since ZŽ G . s 1, a short computation shows that w ␣ , x, x x s 1 for all x g G. This means that ␣ g AŽ G . and proves the claim. To prove the theorem, note that if R 2 Ž G . s 1 then AŽ G . s id G 4 by Lemma 2.4. But then R 2 Ž Aut Ž G .. s id G 4 by the above claim since obviously ZŽ G . s 1.
CONCLUDING REMARKS Ž1. If G is a group, if ␣ is a nontrivial automorphism in AŽ G ., and if we form the semidirect product H s w G x² ␣ :, then it can be shown that ␣ g R 2 Ž H .. Ž2. It is easy to prove that if G is a finite group and if AŽ G . is trivial, then AŽ Aut Ž G .. is also trivial. Indeed, since AŽ G . is trivial, it follows that Autc Ž G . is trivial and since Autc Ž G . is the centralizer of InnŽ G . in Aut Ž G . one obtains that ZŽ Aut Ž G .. is trivial. By a result in w2x which applies to finite groups, one infers that AŽ Aut Ž G .. is trivial. We were not able to prove the corresponding result for arbitrary groups G.
ACKNOWLEDGMENT The authors thank the referee for a number of valuable suggestions.
REFERENCES 1. R. Baer, Engelsche Elemente Noetherscher Gruppen, Math. Ann. 133 Ž1957., 256᎐270. 2. M. Deaconescu, Gh. Silberberg, and G. Walls, On commuting automorphisms of groups, Comm. Algebra, submitted for publication. 3. H. Heineken, Eine Bemerkung ¨ uber engelsche Elemente, Arch. Math. Ž Basel . 11 Ž1960., 321. 4. I. N. Herstein, Problem proposal, Amer. Math. Monthly 91 Ž1984., 203. 5. W. Kappe, Die A-Norm eine Gruppe, Illinois J. Math. 5 Ž1961., 187᎐197. 6. T. J. Laffey, Solution to problem E3039, Amer. Math. Monthly 93 Ž1986., 816. 7. M. F. Newman and W. Nickel, Engel elements in groups, J. Pure Appl. Algebra 96 Ž1994., 39᎐45. 8. M. Pettet, personal communication. 9. D. J. S. Robinson, ‘‘A Course in the Theory of Groups,’’ Springer-Verlag, New YorkrBerlin, 1982.
Right 2-Engel Elements and Commuting Automorphisms of Groups Marian Deaconescu1 Department of Mathematics and Computer Science, Kuwait Uni¨ ersity, P.O. Box 5969, Safat 13060, Kuwait
and Gary L. Walls Department of Mathematics, Uni¨ ersity of Southern Mississippi, Hattiesburg, Mississippi 39401 Communicated by Georgia Benkart Received December 13, 1999
It is shown that there is a close connection between the right 2-Engel elements of a group and the set of the so-called commuting automorphisms of the group. As a consequence, the following general theorem is proved: If G is a group and if R 2 Ž G . denotes the subgroup of right 2-Engel elements, then the factor group X R 2 Ž G . l CG Ž G .rZ2 Ž G . is a group of exponent at most 2. 䊚 2001 Academic Press Key Words: commuting automorphisms; central automorphisms; right 2-Engel elements
INTRODUCTION Let G be a group and consider the set R 2 Ž G . s g g G ¬ w g, x, x x s 1 for all x g G4 of right 2-Engel elements of G. It is well known, see Kappe w5x, that R 2 Ž G . is a characteristic subgroup of G. It is also known, see Heineken w3x, that the inverse of a right 2-Engel element is a left 3-Engel element. 1 While working at this paper, the first named author was supported by Research Grant SM 09r00 from Kuwait University Research Administration.
479 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
480
DEACONESCU AND WALLS
In this paper we are interested in the structure of the subgroup R 2 Ž G . for centerless groups G. Results from w1x and w2x imply that if G is a finite group with trivial center, then R 2 Ž G . s 1. One would expect that whenever G is a group with trivial center R 2 Ž G . is either trivial, or has a structure which is hard to control; indeed, the structure of R 2 Ž G . is known to be under control provided that some rather strong conditions are imposed on G, as, for example, the condition that G satisfies the maximum condition on subgroupsᎏsee, for example, Baer w1x or the discussion in w9, Sect. 12.3x. In a certain sense, centerless groups are as far from being nilpotent as one could get; recent results, see w7x, show that even in nilpotent groups there may exist 5-Engel elements none of whose powers is a left 5-Engel element. The following result comes then as a surprise; it asserts that all groups possess a certain canonic section which has a very special structure: THEOREM 1.1. Let G be a group. Then R 2 Ž G . l CG Ž GX .rZ2 Ž G . is a group of exponent at most 2. In particular, if ZŽ G . s 1, then R 2 Ž G . is a subgroup of G of exponent at most 2. The above theorem is a consequence of results related to the so-called commuting automorphisms of groups. An automorphism ␣ of a group G is called a commuting automorphism if g ␣ Ž g . s ␣ Ž g . g for all g g G. The set of all commuting automorphisms of G is denoted here by AŽ G . and it is clear that set AŽ G . contains the group Autc Ž G . of central automorphisms of G. The converse inclusion does not hold in general; in fact, see w2x, there exist finite nonabelian 2-groups G such that AŽ G . is not a subgroup of Aut Ž G .. Previous results suggest, however, the following: Conjecture. If G is a group and if Autc Ž G . s id G 4 , then AŽ G . s id G 4 . This conjecture was shown to be true in the following particular cases: G is a simple nonabelian group ŽI. N. Herstein w4x., or G has no nontrivial abelian normal subgroups ŽT. J. Laffey w6x., or G s GX and ZŽ G . s 1 ŽM. Pettet w8x., or G is finite ŽDeaconescu et al. w2x.. The second main result of this paper shows that the above conjecture is false: THEOREM 1.2. There exist infinite groups G such that ZŽ G . s 1 and AŽ G . / id G 4 . More information about the groups whose existence is ensured by Theorem 1.2 is contained in THEOREM 1.3.
Let G be a group such that ZŽ G . s 1 and AŽ G . / id G 4 .
Ži. AŽ G . is a subgroup of Aut Ž G . and expŽ AŽ G .. s 2. Žii. 1 - R 2 Ž G . F CG Ž GX . and expŽ R 2 Ž G .. s 2.
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Žiii. If ␣ g AŽ G ., then ␣ fixes G 2 R 2 Ž G ., where G 2 is the subgroup of G generated by the set of squares of elements in G. The above result has an immediate consequence which shows that if a group G has no nontrivial central automorphisms, then the structure of the set AŽ G . is very simple: COROLLARY 1.4. If G is a group such that Autc Ž G . s id G 4 , then AŽ G . is a subgroup of Aut Ž G . of exponent at most 2. It is well known that if G is a group with trivial center, then the center of Aut Ž G . is trivial too. Our last result shows that the same property holds for R 2 Ž G .: If G is a group such that R 2 Ž G . is tri¨ ial, then R 2 Ž Aut Ž G ..
THEOREM 1.5. is tri¨ ial.
PRELIMINARY LEMMAS The first lemma is due to T. J. Laffey w6x and is very useful in computations: LEMMA 2.1.
Let ␣ g AŽ G .. Then w ␣ Ž x ., y x s w x, ␣ Ž y .x for all x, y g G.
The next result bridges together the right 2-Engel elements and the commuting automorphisms of a group: LEMMA 2.2. Ži. If ␣ g AŽ G . and if x g G, then xy1␣ Ž x . g R 2 Ž G .. Žii. Let g g G and denote by Tg the inner automorphism induced by g. Then Tg g AŽ G . if and only if g g R 2 Ž G .. In particular, InnŽ G . l AŽ G . is isomorphic to R 2 Ž G .rZŽ G .. Proof. The proofs of both assertions are elementary and can be found in w2x. The following properties of commuting automorphisms are essential for what follows: LEMMA 2.3. Ži. Žii.
Let ␣ g AŽ G ..
If x g GX , then xy1␣ Ž x . g ZŽ G .. GX : CG Ž ␣ . if and only if ␣ 2 g Autc Ž G ..
Proof. See w2x.
482
DEACONESCU AND WALLS
The following result provides a criterion for AŽ G . to be trivial: LEMMA 2.4. Let G be a group with no nontri¨ ial central automorphisms. Then AŽ G . s id G 4 if and only if R 2 Ž G . s ZŽ G .. In particular, if G s GX , then R 2 Ž G . s ZŽ G .. Proof. If AŽ G . s id G 4 , then R 2 Ž G . s ZŽ G . by Lemma 2.2Žii.. If R 2 Ž G . s ZŽ G ., it follows from Lemma 2.2Ži. that AŽ G . : Autc Ž G ., whence AŽ G . s id G 4 . To prove the last assertion assume that G s GX . Then Autc Ž G . is trivial since central automorphisms of G fix GX elementwise. Lemma 2.3Ži. implies that AŽ G . is trivial. Thus R 2 Ž G . s ZŽ G . by the above paragraph. PROOFS OF THE MAIN RESULTS Proof of Theorem 1.2. We will construct an infinite group G with trivial center and such that R 2 Ž G . is nontrivial. Since ZŽ G . s 1, it follows at once that Autc Ž G . is trivial; the fact that R 2 Ž G . is nontrivial implies, via Lemma 2.4, that AŽ G . is not trivial. Let H denote the direct sum Žrestricted direct product. of copies of the cyclic group of order two, indexed over the set of positive integers. Let G denote the standard restricted wreath product of Z2 by H. That is, G is the semidirect product of the base group B Žwhich is the direct sum of < H < copies of Z2 . by H by using the regular representation of H to define the action of H on B. In this representation each nontrivial element of H acts as a formal product of disjoint 2-cycles which fixes no point in B. Also, B has a Z2-basis Žcorresponding to the elements of H . on which H acts transitively. It is known that any standard wreath product of an abelian group by an infinite group is centerless, thus ZŽ G . is trivial. Let now x s Ž1, 0, 0, . . . . g B and let y s bh be an arbitrary element of G with b g B and h g H. Assuming that h contains the 2-cycle Ž1, i ., then x y s x h s Ž0, 0, . . . , 1, 0, 0, . . . . because 1 and 0 are interchanged and all other interchanges are between zeros. Then w x, y x s Ž1, 0, 0, 0, . . . , 1, 0, 0, . . . . and y commutes with w x, y x since b does and since h will interchange the two 1’s and a set of zeros. It follows that Ž1, 0, 0, . . . . is a right 2-Engel element, as are all tuples with only one nonzero component. But these elements generate B. Thus B is contained in R 2 Ž G . and since R 2 Ž G . is abelian it follows that B s R 2 Ž G .. Thus here R 2 Ž G . is an infinite group of exponent 2 and this completes the proof of Theorem 2.1. Remark. The above considerations show that all the elements with exactly two 1’s are in GX . These elements generate the subgroup of B of elements which have an even number of 1’s. That is clearly a normal
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subgroup of G with abelian quotient. So GX is this subgroup and therefore GX centralizes B s R 2 Ž G .. This last property of our example is not accidental; it will turn out that R 2 Ž G . centralizes GX in every group G with trivial center. Proof of Theorem 1.3. Since ZŽ G . s 1, it follows at once that Autc Ž G . is trivial. By hypothesis, AŽ G . is nontrivial, so by Lemma 2.4, R 2 Ž G . / 1. Fix a nontrivial automorphism ␣ g AŽ G .. Since ZŽ G . s 1, Lemma 2.3Ži. implies that ␣ fixes GX and by Lemma 2.3Žii. one concludes that ␣ 2 s id G . This shows that expŽ AŽ G .. s 2. For x g G, we have that xy1␣ Ž x . g R 2 Ž G . by Lemma 2.2Ži.. Now since Txy1 ␣ Ž x . g AŽ G . by Lemma 2.2Žii., one obtains that the order of this inner automorphism is at most 2. Hence Ž xy1␣ Ž x .. 2 s 1 and since x ␣ Ž x . s ␣ Ž x . x it follows that ␣ Ž x 2 . s x 2 . Thus the subgroup G 2 s ² g 2 ¬ g g G4: is fixed by ␣ . Take now r g R 2 Ž G ., so that Tr g AŽ G . by Lemma 2.2Žii. and Tr2 s id G . Then r 2 g ZŽ G . s 1 and we obtain that expŽ R 2 Ž G .. s 2. Moreover, Lemma 2.3Žii. implies that Tr fixes GX , whence r g CG Ž GX .. Hence R 2 Ž G . F CG Ž GX . and the proof of part Žii. of the theorem is complete. To finish the proof of part Ži., note that AŽ G . is closed with respect to taking inverses. Let ␣ ,  g AŽ G . and let x g G. Then, since xy1␣ Ž x ., xy1 Ž x . g R 2 Ž G . and since R 2 Ž G . is abelian, we derive, using Lemma 2.1, that 1 s w xy1␣ Ž x ., xy1 Ž x .x s w ␣ Ž x .,  Ž x .x s w x, ␣ Ž x .x, which shows that ␣ g AŽ G .. Thus AŽ G . is a subgroup of Aut Ž G ., which proves Ži.. To complete the proof of part Žiii., we only need to show that every ␣ g AŽ G . fixes R 2 Ž G .. Take r g R 2 Ž G ., so that the inner automorphism Tr g AŽ G .. Since by part Ži., AŽ G . is abelian, it follows that w Tr , ␣ x s id G . Since ZŽ G . s 1, a short calculation shows that ␣ Ž r . s r. Proof of Theorem 1.1. Let r g R 2 Ž G . l CG Ž GX .. Then the inner automorphism Tr is a commuting automorphism of G which fixes GX , so by Lemma 2.3Žii., Tr2 g Autc Ž G . l InnŽ G .. But then, since Autc Ž G . l InnŽ G . s ZŽ InnŽ G .. ( Z2 Ž G .rZŽ G ., it follows at once that r 2 g Z2 Ž G .. This proves the first assertion of Theorem 1.1. If ZŽ G . s 1, then Z2 Ž G . s 1 and by Theorem 1.3Žii., R 2 Ž G . F CG Ž GX .. Thus R 2 Ž G . ( R 2 Ž G . l CG Ž GX .rZŽ G . is a group of exponent at most 2 by the first part of the theorem. The proof is now complete. Proof of Corollary 1.4. If G has no nontrivial central automorphisms, then necessarily Z2 Ž G . s ZŽ G .. Set G s GrZŽ G . and observe that every ␣ g AŽ G . induces an automorphism ␣ g AŽ G .. But since G has trivial center, it follows from Theorem 1.3Ži. that ␣ 2 s id G . Now the function : Aut Ž G . ª Aut Ž G ., defined by Ž ␣ . s ␣ , is one-by-one since Ker s Autc Ž G . s id G 4 . Thus AŽ G . is embedded into AŽ G . and the result follows from Theorem 1.3Ži..
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Proof of Theorem 1.5. We claim first that if G is a group with ZŽ G . s 1, then R 2 Ž Aut Ž G .. : AŽ G .. Indeed, let ␣ g R 2 Ž Aut Ž G ... Then w ␣ , Tx , Tx x s id G for every Tx g InnŽ G .. Since ZŽ G . s 1, a short computation shows that w ␣ , x, x x s 1 for all x g G. This means that ␣ g AŽ G . and proves the claim. To prove the theorem, note that if R 2 Ž G . s 1 then AŽ G . s id G 4 by Lemma 2.4. But then R 2 Ž Aut Ž G .. s id G 4 by the above claim since obviously ZŽ G . s 1.
CONCLUDING REMARKS Ž1. If G is a group, if ␣ is a nontrivial automorphism in AŽ G ., and if we form the semidirect product H s w G x² ␣ :, then it can be shown that ␣ g R 2 Ž H .. Ž2. It is easy to prove that if G is a finite group and if AŽ G . is trivial, then AŽ Aut Ž G .. is also trivial. Indeed, since AŽ G . is trivial, it follows that Autc Ž G . is trivial and since Autc Ž G . is the centralizer of InnŽ G . in Aut Ž G . one obtains that ZŽ Aut Ž G .. is trivial. By a result in w2x which applies to finite groups, one infers that AŽ Aut Ž G .. is trivial. We were not able to prove the corresponding result for arbitrary groups G.
ACKNOWLEDGMENT The authors thank the referee for a number of valuable suggestions.
REFERENCES 1. R. Baer, Engelsche Elemente Noetherscher Gruppen, Math. Ann. 133 Ž1957., 256᎐270. 2. M. Deaconescu, Gh. Silberberg, and G. Walls, On commuting automorphisms of groups, Comm. Algebra, submitted for publication. 3. H. Heineken, Eine Bemerkung ¨ uber engelsche Elemente, Arch. Math. Ž Basel . 11 Ž1960., 321. 4. I. N. Herstein, Problem proposal, Amer. Math. Monthly 91 Ž1984., 203. 5. W. Kappe, Die A-Norm eine Gruppe, Illinois J. Math. 5 Ž1961., 187᎐197. 6. T. J. Laffey, Solution to problem E3039, Amer. Math. Monthly 93 Ž1986., 816. 7. M. F. Newman and W. Nickel, Engel elements in groups, J. Pure Appl. Algebra 96 Ž1994., 39᎐45. 8. M. Pettet, personal communication. 9. D. J. S. Robinson, ‘‘A Course in the Theory of Groups,’’ Springer-Verlag, New YorkrBerlin, 1982.