On Marginal Automorphisms of a Group Fixing the Certain Subgroup

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1 Global Analysis and Discrete Mathematics Volume 2, Issue 2, pp ISSN: On Marginal Automorphisms of a Group Fixing the Certain Subgroup Mohammad Mehdi Nasrabadi Samane Hajizade Received: 16 March 2017 / Accepted: 16 July 2017 Abstract Let W be a variety of groups defined by a set W of laws and G be a finite p-group in W. The automorphism α of a group G is said to be a marginal automorphism (with respect to W ), if for all x G, x 1 α(x) W (G), where W (G) is the marginal subgroup of G. Let M, N be two normal subgroups of G. By Aut M (G), we mean the subgroup of Aut(G) consisting of all automorphisms which centralize G/M. Aut N (G) is used to show the subgroup of Aut(G) consisting of all automorphisms which centralize N. We denote Aut N (G) Aut M (G) by Aut M N (G). In this paper, we obtain a necessary and sufficient condition that Aut w (G) = Aut W (G) W (G) (G). Keywords W-nilpotent group marginal automorphism purely non-abelian group Mathematics Subject Classification (2010) 20D45 20E36 Mohammad Mehdi Nasrabadi Department of Mathematics, University of Birjand, Iran. Tel.: Fax: mnasrabadi@birjand.ac.ir Corresponding author Samane Hajizade Department of Mathematics, University of Birjand, Iran. hajizadehmath@gmail.com c 2017 Damghan University. All rights reserved.

2 136 Mohammad Mehdi Nasrabadi, Samane Hajizade 1 Introduction 2 Introduction and results Let G be a group and F be the free group on the countable set {x 1, x 2, } and W be a non-empty subset of F. Suppose that W (G) = w(g 1,, g r ) w = w(x 1,, x r ) W and g 1,, g r G and { W (G) = g G w(g 1,, g i 1, g i g, g i+1,, g r ) = w(g 1,, g r ) } for all w W, for all g 1,, g r G and for all 1 i r. For a group G, W (G) and W (G) are called, respectively, the verbal subgroup and the marginal subgroup of G with respect to W (see [8]). The verbal subgroup W (G) is a fully invariant subgroup of G, and the marginal subgroup W (G) is a characteristic subgroup of G. We call an automorphism α of G a marginal automorphism with respect to W if x 1 α(x) W (G) for all x G. The set of all marginal automorphisms of G forms a normal subgroup Aut w (G) of the automorphism group Aut(G) of G. If we take W = {[x 1, x 2 ]} where [x 1, x 2 ] = x1 1 x 1 2 x 1x 2, then W (G) = G and W (G) = Z(G) where G and Z(G) are the commutator subgroup and the center of G, respectively. In this case, Aut w (G) is denoted by Aut c (G) and the elements of Aut c (G) are called central automorphism of G. There are some well-known results about central automorphisms of G ( for example see [1], [6] and [7] ). Let M, N be two normal subgroups of G. By Aut M (G), we mean the subgroup of Aut(G) consisting of all automorphisms which centralize G/M. Aut N (G) is used to show the subgroup of Aut(G) consisting of all automorphisms which centralize N. We denote Aut N (G) Aut M (G) by Aut M N (G). Let be W a variety of groups defined by set W of laws then a group G is said to be W-nilpotent if there exist a series 1 = G 0 G 1 C k = G (1) such that G i G and G i+1 /G i W (G/G i ), for 0 i n. The length of the shortest series (1) is the W-nilpotent class of G. For a finite p-group G define Ω 1 (G) = x G x p = 1. A non-abelian group G is called purely non-abelian if it has no non-trivial abelian direct factor. In this article such that group showed by PN-group. For any group H and abelian group K, Hom(H, K) denote the group of all homomorphisms from H to K. Azhdari and Malayeri [4] found necessary and sufficient condition that Aut M N (G) = Aut c(g) and several corollaries where M, N are normal subgroups of a finite p-group G. Through this article, W is a variety of groups and G is a finite p-group in

3 On Marginal Automorphisms of a Group Fixing the Certain Subgroup 137 W which W (G) Z(G) and G/W (G) is abelian also by the assumption M W (G) we have : M = C p a 1 C p a 2 C p as (2) W (G) = C p b 1 C p b 2 C p b s (3) G/W (G) = C p d 1 C p d 2 C p dr (4) where a 1 a 2 a s > 0, b 1 b 2 b s > 0 and also d 1 d 2 d r > 0 Let t be the smallest integer between 1 and s such that a j = b j for all t s and t + 1 j s. By this assumption our main results are the following. Main theorem: Let W be a variety of group, G be finite p-group in W which is W-nilpotent. Let M 1, M 2, and be normal subgroups of G such that M i W (G) N i for i = 1, 2, M 1 M 2 and. Then Aut M 1 (G) if and only if one of the following statements holds: (i) M 1 = M 2 and W (G)G pn where exp(m 1 ) = p n or (ii) =, s = s and exp(g/w (G) ) p at, where t is the smallest integer between 1 and s such that a j = b j. By using the above notation we have the following corollary. Corollary. Let G be a finite p-group which is W-nilpotent group. Let M and N be two normal subgroups of G such that M W (G) W (G). Then (i) Aut M N (G) = AutW (G) W (G) (G) if and only if one of the following statements holds: M = W (G) and N W (G)G pn W (G) where exp(w (G)) p a n or N = W (G), s = s and exp(g/w (G)W (G)) p at ; (ii) Aut M N (G) = Aut w (G) if and only if one of the following statements holds: M = W (G) and N W (G)G pn where exp(w (G)) = p an or N W (G), s = s and exp(g/w (G)) p a t. (5) 3 Preliminary results Adeney and Yen [1, Theorem 1] prove that if G is a purely non-abelian, then there exist a bijection between Aut c (G) and Hom(G/G, Z(G)). Also Jamali and Mousavi in [6] prove that if G is a finite group such that Z(G) G then Aut c (G) = Hom(G/G, Z(G)). Similarly Attar in [3] prove the following theorems about marginal automorphisms of a group G:

4 138 Mohammad Mehdi Nasrabadi, Samane Hajizade Theorem 1 [3] Let G be a group and W F be the set of laws, Then Aut w (G) acts trivially on W (G). Theorem 2 [3] Let G be a purely non-abelian finite group and W F be the set of laws such that W (G) Z(G), then Aut w (G) = Hom ( G/W (G), W (G) ). Theorem 3 [3] Let G be a group and W F be the set of laws such that W (G) Z(G) W (G), Then Aut w (G) = Hom ( G/W (G), W (G) ) Proposition 1 [3] Let G be a purely non-abelian finite group and W F be the set of laws such that W (G) is abelian, Then (1) For each α Hom ( G, W (G) ) and t W (G) we have α(t) = 1; (2) Hom ( G/W (G), W (G) ) = Hom ( G, W (G) ). By this proposition we have Aut w (G) = Aut W (G) W (G) (G) We recall that through this article W is a variety of groups and G is a finite p-group in W which W (G) Z(G) and G/W (G) is abelian with abelian direct factor (4). Theorem 4 [3] Let W is a variety of groups and G is W-nilpotent group and 1 N G, then N W (G) 1. Lemma 1 [3] Let G be a finite PN-group and M, N be two normal subgroups of G such that M Z(G)( in particular M W (G) since W (G) Z(G)), then Aut M N (G) = Hom ( G/N, M ). Lemma 2 [3] Let G be a group and M, N be two normal subgroups of G such that M Z(G) N, then and Aut M N (G) is abelian. Aut M N (G) = Hom ( G/N, M ) Lemma 3 [9] Let A and B be two finite abelian p-groups such that A = C p a 1 C p a 2 C p as where a 1 a 2 a s > 0 and B = C p b 1 C p b 2 C p bs where b 1 b 2 b s > 0. Let b j a j for all j, 1 j s and b j > a j for some such j. Let t be the smallest integer between 1 and s such that a j = b j for all j such that t + 1 j s. Then for any finite abelian p-group C, Hom(A, C) < Hom(B, C) if and only if the exponent of C is at least p at+1.

5 On Marginal Automorphisms of a Group Fixing the Certain Subgroup Proof of the main results Proof of The Main Theorem: We proceed with a series of steps. Let G be a finite p-group which is W-nilpotent. Step 1. Let M W (G) and exp(m) = p n and N be a normal subgroup of G. Then for all f Hom(G/N, M) we have kerf = W (G)G pn N/N. Proof. Clearly W (G)G pn N/N kerf for all f Hom(G/N, M). To prove the converse inclusion, let x W (G)G pn N. Since M W (G), then Hom(G/N, M) = Hom(G/W (G)N, M). Put Ḡ = G/W (G)N. Ḡ is a finite abelian p-group and so there exist x 1, x 2,, x t G such that Ḡ = x 1 x 2 x t and xw (G)N = x ps 1 1 x ps t t W (G)N for suitable s i 0 (See [7 lemma 2.2] ) Since x W (G)G pn N/W (G)N, x ps j j G pn for some 1 j t and therefor s j < n. Now choose element z M such that z = min{ x j, p n }, and define a homomorphism f z : x j z from Ḡ to M. When M W (G) and K is a direct factor of Ḡ then any element f of Hom( K, M) induces an element f of Hom(Ḡ, M) which is trivial on the complement of K of Ḡ. To simplify the notion, we will identify f with the corresponding homomorphism from Ḡ to M. We have f p z( x) = f z ( x s j j ) = z ps j 1. Thus x kerf for f Hom(G/N, M) and consequently the equality holds. Step 2. Let, be two normal subgroups of G such that and M W (G) N i for i = 1, 2. Then Aut M = Aut M if and only if W (G)G pn where exp(m) = p n. Proof. Since, Aut M Aut M. Suppose that W (G)G pn, by using Step 1, N1 kerf for all f Hom(G/, M). We have Hom(G/, M) = Hom(G/, M) = Hom(G/, M), since. That is Aut M (G) = Aut M (G) and hence Aut M (G) = Aut M (G). Conversely, assume that Aut M (G) = Aut M (G). Then α(n) = n for all n and α Aut M (G). By Lemma 2 α ( n) = 1 for all α Hom(G/, M) and n. Consequently by Step 1, W (G)G pn, as required. Step 3. Let N be a normal subgroup of G and M 1 < M 2 W (G)G pn N. Then Aut M 1 N (G) = AutM 2 N (G) if and only if s = s and exp(g/w (G)N) p a t where t is the smallest integer between 1 and s such that a j = b j for all t+1 j s. Proof. Since M 1 M 2, Aut M 1 N (G) AutM 2 N (G) by using Lemma 2 AutM 1 N (G) = Aut M 2 N (G) if and only if Hom(G/N, M 1) = Hom(G/N, M 2 ). first assume that Aut M 1 N (G) = AutM 2 N (G), then clearly s = s. So by applying Lemma 3 with A = M 1, B = M 2 and C = W (G)N we get exp(g/w (G)N) p a t, since if exp(g/w (G)N) p at+1, then we have Hom(G/N, M 1 ) < Hom(G/N, M 2 ) which is a contradiction. Now suppose that s = s and exp(g/w (G)N) p a t then Hom(G/N, M 1 ) = Hom(G/N, M 2 ) and thereforaut M 1 N (G) AutM 2 N (G). Step 4. First assume that Aut M 1 (G). By Lemma 2 Hom(G/, M 1 ) = Hom(G/, M 2 ). If M 1 M 2 and then by Lemma D [5] Hom(G/, M 1 ) Hom(G/, M 2 ). This contradiction implies M 1 = M 2 or =. If M 1 = M 2 then by Step 2, W (G)G pn. Else, since M 1 M 2, = then by Step 3, it follows that s = s and

6 140 Mohammad Mehdi Nasrabadi, Samane Hajizade exp(g/w (G)N) p at. Conversly, if (i) or (ii) holds it is easy to see that Hom(G/, M 1 ) = Hom(G/, M 2 ). On the other hand since M 1 M 2 and, Aut M 1 (G) Aut M 2 (G) and consequently Aut M1 (G) = Aut M2 (G), as required. Remark. Note that in the Proof of The Main Theorem, we use the conditions M i W (G) N i only to prove the equality Aut M i N i = Hom(G/N i, M). So by Lemma 1 we may substitute this condition by G be a PN-group. And by using the same argument, we can easily prove the following Theorem. Theorem 5 Let G be a finite p-group which is PN and W-nilpotent group. Let M 1 and M 2 be two subgroups which is marginal subgroup and and be two normal subgroups of G such that M 1 M 2 and. Then Aut M 1 (G) Aut M 2 (G) if and only if one of the following statements holds: (i) M 1 = M 2 and W (G)G pn where exp(m 1 ) p n, or (ii) =, s = s and exp(g/w (G) ) p a t where t is the smallest integer between 1 and s such that a j = b j for all t + 1 j s. The Main Theorem has a number of important consequences. As a first application of this we get the following result. Corollary 1 Let G be a finite p-group which is W-nilpotent group. Let M and N be two normal subgroups of G such that M W (G) W (G). Then (i) Aut M N (G) = AutW (G) W (G) (G) if and only if one of the following statements holds: M = W (G) and N W (G)G pn W (G) where exp(w (G)) p a n or N = W (G), s = s and exp(g/w (G)W (G)) p at ; (ii) Aut M N (G) = Aut w (G) if and only if one of the following statements holds: M = W (G) and N W (G)G pn where exp(w (G)) = p an or N W (G), s = s and exp(g/w (G)) p a t. Proof (i) The result follows immediately by applying The Main Theorem with M 1 = M, = N and M 2 = = W (G). To prove (ii), note that if M contained in the marginal subgroup of G, then Aut M N (G) = AutM NW (G) (G). Also M W (G) N follows that Aut w (G) = Aut W (G) W (G) (G) and since G is a PN-group therefor, by applying Theorem 5 with M 1 = M, = NW (G), M 2 = W (G) and = W (G), (ii) holds. Conversely, if the firs part of (ii) holds then Hom ( G/N, M ) ( = Hom G/N, W (G) ) ( = Hom G/W (G)N, W (G) ) = Hom ( G/W (G), W (G) )

7 On Marginal Automorphisms of a Group Fixing the Certain Subgroup 141 Now if the second part of (ii) holds then Hom ( G/N, M) = Hom ( G/NW (G), W (G) ) = Hom ( G/N, W (G) ) hence Hom ( G/N, M ) = Hom ( G/W (G), W (G) ) or equivalently Aut M N (G) = Aut w (G). This corollary yields some following results. Corollary 2 Let G be a finite p-group which is W-nilpotent group. Then Aut w (G) = Aut W (G) W (G) (G) if and only if W (G) W (G)G pn where exp ( W (G) ) = p n. Proof It follows from Corollary 1 when M = N = W (G). Corollary 3 Let G be a finite p-group which is W-nilpotent group. Let M 1, M 2, and be normal subgroups of G such that M i W (G) N i for i = 1, 2. Then Aut M1 (G) = Aut M2 (G) if and only if one of the following statements holds: (i) M 1 = M 2 and N i W (G)G pn j N j for i = 1, 2 and i j; (ii) M 1 M 2,s = s 1 = s 2, W (G)G pn 1 and exp ( ) G/W (G) p at 2 ; (iii) M 2 M 1, s = s 1 = s 2, W (G)G pn 2 and exp ( ) G/W (G) p at 1 ; (iv) =, s = s 1 = s 2 and exp ( ) G/W (G) p for i = 1, 2 Proof First assume that Aut M1 (G) = Aut M2 (G). Therefore we have Aut M1 (G) = Aut M 1 M 2 (G). Clearly, M 1 M 2 M i and N i for i = 1, 2 and so we may apply The Main Theorem. Since Aut Mi N i (G) = Aut M 1 M 2 (G) for i = 1, 2 one of the following case happens. (I) M i = M 1 M 2 and W (G)G pn i N i. So M i M j and N j W (G)G pn i N i for i j. Or, (II) N i =, s = s i and exp ( ) G/W (G)N i p. So N j N i, s = s i and exp ( ) G/W (G)N i p for i j. Therefore we have the following four cases: (1) If for i = 1, 2 (I) holds, then M 1 = M 2 and N i W (G)G pn j N j for i, j = 1, 2 and i j and hence (i) follows. (2) If for i = 1 (I) and for i = 2 (II) happen, then M 1 M 2, W (G)G pn 1 and so s = s 2, and exp ( ) G/W (G) p a t 2. Since a t 2 n 1, G pn 1 W (G) and implies that W (G) = W (G)G pn 1. Furthermore from M 1 M 2, it follows that s = s 1 and consequently, M 1 M 2, s = s 1 = s 2, W (G)G pn 1 and exp ( ) G/W (G) p a t 2 and so in this case (ii) holds.

8 142 Mohammad Mehdi Nasrabadi, Samane Hajizade (3) If for i = 1 (II) and for i = 2 (I) happen, then with the argument similar to the case (2) we may conclude (ii) holds. (4) Finally if i = 1, 2 (II) holds, then evidently = =, s = s 1 = s 2 and exp ( ) G/W (G) p, where i = 1, 2 that is (iv). Conversely. First assume that (i) holds. Then we have M 1 = M 2 = M and from N i W (G)G pn j N j. It follows that Hom(G/N j, M) = Hom(G/, M) for i, j = 1, 2 and i j. Consequently Aut M N i (G) = Aut M (G) and hence Aut M (G) = Aut M (G). Now suppose that (ii) holds. Since =, s = s 2 and exp ( ) G/W (G) p at 2, we have Aut M 1 M 2 (G). Also W (G)G pn 1 concludes that Aut M1 (G) = Aut M1 (G) = Aut M1 M2 (G) since M 1 M 2. Therefore Aut M 1 (G). The case (ii) follows, by a similar argument. Finally suppose that (iv) holds. So = = N, s = s 1 = s 2 and exp ( ) G/W (G)N i p for i = 1, 2, it follows that Aut M i N (G) = AutM 1 M 2 N (G) and this completes the proof. Note that here also, the condition M i W (G) N i for i = 1, 2 can be replaced by condition G be a PN-group. Corollary 4 Let G be a finite p-group which is PN and W-nilpotent group. Let M 1, M 2, and be two normal subgroups of G such that M i W (G) for i = 1, 2. Then Aut M1 (G) = Aut M2 (G) if and only if one of the following statements holds: (i) M 1 = M 2 and N i W (G)G pn j N j for i = 1, 2 and i j; (ii) M 1 M 2,s = s 1 = s 2, W (G)G pn 1 and exp ( ) G/W (G) p at 2 ; (iii) M 2 M 1, s = s 1 = s 2, W (G)G pn 2 and exp ( ) G/W (G) p at 1 ; (iv) =, s = s 1 = s 2 and exp ( ) G/W (G) p for i = 1, 2 Another interesting equality is indicated by the following result. Theorem 6 Let G be a finite p-group which is PN and W-nilpotent group. Let M,, and be normal subgroups of G such that M W (G). If the invariants of M (in the cyclic decomposition) are greather than or equal to exp ( ) G/W (G)N i for i = 1, 2 then Aut M 1 (G) if and only if W (G) = W (G). Proof Let M = C p a 1 C p a 2 C p as and exp ( G/W (G)N i ) = p n i for i = 1, 2. First assume that. By assumption a i a j for all 1 j s and i = 1, 2. Consequently we have Hom(G/N i, M) = Hom(G/N i, C p a 1 ) Hom(G/N i, C p as ) = (G/W (G)Ni ) n.

9 On Marginal Automorphisms of a Group Fixing the Certain Subgroup 143 Therefore Aut M (G) = Aut M (G) if and only if G/W (G) = G/W (G) or equivalently W (G) = W (G). Since Aut M (G) = Aut M (G) if and only if Aut M N i (G) = Aut M (G) for i = 1, 2, the general case follows. References 1. J.E. Adney and T. Yen,Automorphisms of a p-group, Illinois J. Math, 9, (1965). 2. M.S. Attar, Finite p-group in which each central automorphism fixes center element-wise, Communication in Algebra, 40, (2012). 3. M.S. Attar, On the marginal automorphisms of a group, Comm. Algebra, 37, no 7, (2009). 4. A.Z. Azhdari, M.A. Malayeri, On automorphisms fixing certain groups, J. Algebra Appl, 12, (2013). 5. M.J. Curran, D.J. McCaughan, Central automorphisms that are almost inner, Commun. Algebra, 29(5), (2001). 6. A. Jamali, H. Mousavi, On the central automorphisms group of finite p-groups, Algebra Colloquium, 9, 7 14 (2002). 7. S.H. Jaffari, Central automorphism grroups fixing the center element-wise, Ent. Electron. J. Aljebra, 9, (2011). 8. S.M. Tehrani, F. Shahini, Varietal nilpotent groups, World Applied Science Journal, 12, (2012). 9. M.K. Yadave, On central automorphisms fixing the center ele7ment-wise, Commun. Aljebra, 37, (2009).