aa r X i v : . [ m a t h . G R ] J u l ON SUBGROUPS GENERATED BY SMALL CLASSES INFINITE GROUPS
MANOJ K. YADAV
Abstract.
Let G be a finite group and M ( G ) be the subgroup of G generatedby all non-central elements of G that lie in the conjugacy classes of the smallestsize. Recently several results have been proved regarding the nilpotency classof M ( G ) and F ( M ( G )), where F ( M ( G )) denotes the Fitting subgroup of M ( G ). We prove some conditional results regarding the nilpotency class of M ( G ). Introduction
Let G be an arbitrary finite group. An element x of G is said to be small if x isnon-central and lies in some conjugacy class of the smallest size in G . Let M ( G )denote the subgroup of G which is generated by all small elements of G .For a given subgroup H of G and an element x ∈ G , by x H we denote the entire H -class of x constisting of elements of the form h − xh , where h runs over everyelement of H . For the same H and x , we denote the set { [ x, h ] | h ∈ H } by [ x, H ],where [ x, h ] denotes the commutator x − h − xh of x and h . Since x H = x [ x, H ], itfollows that | x H | = | [ x, H ] | . By C H ( x ) we denote the centralizer of x in H . A subset S of a group G is said to be normal if g − Sg = S (or equivalently g − Sg ⊆ S ) forall g ∈ G . We write the subgroups in the lower central series of G as γ n ( G ), where γ ( G ) = G and γ n +1 ( G ) = [ γ n ( G ) , G ] for all n > G is a finite group in which the conjugacyclasses of non-central elements are all of the same size n (say), then G must benilpotent and n = p m for some prime integer p and some positive integer m . In2002, K. Ishikawa [2] proved that the nilpotency class of these groups can not bemore than 3. In 2006, A. Mann [4] proved that for any finite nilpotent group G , M ( G ) is nilpotent of class at most 3. Recently M. Isaacs [1, Theorem A] generalizedthese results for non-nilpotent groups. He proved that if a finite group G containsa normal abelian subgroup A such that C G ( A ) = A , then M ( G ) is nilpotent, and ithas nilpotency class at most 3. Isaacs’ result shows that if G is a finite group whichis either nilpotent or supersolvable, then M ( G ) is nilpotent of class at most 3. Incontinuation, A. Mann [5] proved that M ( G ) is nilpotent of class at most 3 if either M ( G ) is solvable and contains a normal subgroup N with abelian Sylow subgroupssuch that G/N is nilpotent, or G is solvable and contains a normal subgroup N with abelian Sylow subgroups such that G/N is supersolvable.The aim of this note is to prove some conditional results regarding the nilpotencyclass of M ( G ) for a finite group G and to give a further research direction to the Mathematics Subject Classification.
Primary 20D25.
Key words and phrases.
Conjugacy class, Fitting subgroup, solvable, nilpotency class.
1N SUBGROUPS GENERATED BY SMALL CLASSES IN FINITE GROUPS 2 topic. Let F ( G ) denote the Fitting subgroup of a given finite group G . Then weprove the following theorem. Theorem A.
Let G be a finite group such that C G ( F ( G )) ≤ F ( G ) and [ x, F ( G )] isa normal subset of F ( G ) for every small element x of G . Then M ( G ) is nilpotentof class at most . To elaborate Theorem A, we would like to remark that the conditions givenin this theorem are naturally satisfied in many cases. Some of such instances arementioned below in Theorem B. The following conjecture is posed by AlexanderMoreto (private communication):
Conjecture A.
Let G be a finite solvable group with trivial center. Then everysmall element of G lies in the center of F ( G ).In Proposition 2.5, we prove that Conjecture A is equivalent to the followingconjecture: Conjecture B.
Let G be a finite solvable group with trivial center. Then, forevery small element x of G , [ x, F ( G )] is a normal subset of F ( G ).Following [6], we say that a subgroup N of a group G is c -closed if any twoelements of N , which are conjugate in G , are also conjugate in N . More results onthis topic can be found in [7]. Following [8], we say that a group G is flat if [ x, G ]is a subgroup of G for all x ∈ G . The following theorem elaborates the usefulnessof Conjecture B. Theorem B.
Let G be a finite solvable group with trivial center. Then ConjectureA holds true if one of the following holds: (1) F ( G ) is ablein; (2) γ ( G ) ∩ F ( G ) ≤ Z( F ( G )) ; (3) F ( G ) is c -closed in G and the nilpotency class of F ( G ) is ; (4) M ( G ) ≤ F( G ) and F ( G ) is flat. Notice that, in all given cases of Theorem B, [ x, F ( G )] is a normal subset of F ( G )for every small element x of G . Thus Conjecture B holds true. Now the proof ofTheorem B follows from the equivalence of Conjectures A and B. We would liketo remark that Conjcture A can also be stated in the following equivalent form,which can be proved easily by Proposition 2.3 given below: Conjecture C.
Let G be a finite solvable group with trivial center. Then, forevery small element x of G , [ x, F ( G )] ⊆ Z( F ( G )).In the following theorem, we generalize (at least in principle) Isaacs’ result [1,Theorem A]. Theorem C.
Let G be a finite group that contains a normal subgroup A such that C G ( A ) ≤ A and [ A, x ] is a normal subset of A for all small elements x ∈ G . Then M ( G ) is nilpotent, and it has nilpotency class at most . Notice that [
A, x ] is a normal subset of A if it is contained in Z( A ). ThusTheorem A gives the following result, which in turn readily gives Isaacs’ result. N SUBGROUPS GENERATED BY SMALL CLASSES IN FINITE GROUPS 3
Corollary D.
Let G be a finite group that contains a normal subgroup A such that C G ( A ) ≤ A and [ A, x ] is contained in the center of A for every small element x of G . Then M ( G ) is nilpotent of class at most . Proofs
We start with the following lemma which is a generalization of Lemma 1 ofMartin Isaacs [1].
Lemma 2.1.
Let G be a finite group and K be a normal subgroup of G . Let x ∈ G − Z( G ) such that [ x, K ] is a normal subset of K . Then, for any y ∈ [ x, K ] , | C G ( y ) | > | C G ( x ) | .Proof. If y = 1, then the result follows trivially, since x is non-central element of G . So assume that y = 1. Now let H = K C G ( x ). Notice that y ∈ H , | x K | = | x H | and C H ( x ) = C G ( x ). Therefore it suffices to show that | H : C H ( y ) | < | H : C H ( x ) | .We claim that [ x, K ] is a normal subset of H . For, let h ∈ H and [ x, k ] ∈ [ x, K ]. Then h = uv , for some u ∈ K and v ∈ C G ( x ). So w := h − [ x, k ] h = v − u − [ x, k ] uv = v − [ x, k ] v for some k ∈ K , since [ x, K ] is a normal subset of K . Now w = [ v − xv, v − kv ] = [ x, v − kv ] = [ x, k ] for k = v − kv ∈ K , since v ∈ C G ( x ) and K is normal in G . Thus w ∈ [ x, K ]. This proves our claim, i.e.,[ x, K ] is a normal subset of H . Notice that 1 = [ x, ∈ [ x, K ]. Since y is anon-trivial element of [ x, K ], which is normal in H , it follows that y H consists ofnon-identity elements of [ x, K ]. Thus | y H | < | [ x, K ] | . Since | [ x, K ] | = | x K | and | x K | = | x H | , we get | H : C H ( y ) | = | y H | < | [ x, K ] | = | x K | = | x H | = | H : C H ( x ) | . This completes the proof of the lemma. (cid:3)
As an application of Lemma 2.1 we have the following two propositions.
Proposition 2.2.
Let K be a normal subgroup of an arbitrary finite group G such that [ x, K ] is a normal subset of K for all small elements x of G . Then [ M ( G ) , K ] ≤ Z( G ) .Proof. It suffices to prove that [ x, K ] ⊆ Z( G ) for all small elements x of G . Let x be an arbitrary small element of G such that [ x, K ] is normal in K as a subset.Let y ∈ [ x, K ] be an arbitrary element. Then it follows from Lemma 2.1 that | y G | < | x G | . Hence | y G | = 1, and hence y ∈ Z( G ). (cid:3) Proposition 2.3.
Let K be a normal subgroup of an arbitrary finite group G and x be a small element of G . Then [ x, K ] ⊆ Z( K ) if and only if [ x, K ] ⊆ Z( G ) .Proof. Suppose that [ x, K ] ⊆ Z( G ). Since K is normal in G , [ x, K ] ⊆ K . Thus[ x, K ] ⊆ K ∩ Z( G ) ≤ Z( K ). Conversely, suppose that [ x, K ] ⊆ Z( K ). Then [ x, K ]is a normal subset of K . Let y ∈ [ x, K ] be an arbitrary element. Then it followsfrom Lemma 2.1 that | y G | < | x G | . Hence | y G | = 1, and hence y ∈ Z( G ). (cid:3) Now we are ready to prove our results stated in the introduction.
N SUBGROUPS GENERATED BY SMALL CLASSES IN FINITE GROUPS 4
Proof of Theorem A.
By the given hypothesis, we have C G ( F ( G )) ≤ F ( G ). Let¯ G = G/ Z( G ). We claim that C ¯ G ( F ( ¯ G )) ≤ F ( ¯ G ). Let π be the natural projectionfrom G to ¯ G . Let C be the inverse image of C ¯ G ( F ( ¯ G )) = C ¯ G ( F ( G ) / Z( G )) under π .Then it follows that C is a normal subgroup of G containing C G ( F ( G )) and satifying[ C, F ( G )] ≤ Z( G ). Now ˆ C = C/ C G ( F ( G )) acts faithfully as automorphisms of F ( G ) centralizing both Z( G ) and F ( G ) / Z( G ). It follows that ˆ C is abelian, andacts on the abelian group C G ( F ( G )) = Z( F ( G )). Furthermore, it centralizes bothZ( F ( G )) / Z( G ) and Z( G ). This implies that C is nilpotent as well as normal in G . Therefore C ≤ F ( G ) and C/ Z( G ) = C ¯ G ( F ( ¯ G )) ≤ F ( G ) / Z( G ) = F ( ¯ G ). Thisproves our claim.Let x be an arbitrary small element of G . Then by the given hypothesis weknow that [ x, F ( G )] is a normal subset of F ( G ). Thus it follows from the proofof Proposition 2.2 that [ x, F ( G )] ⊆ Z( G ). So the image ¯ x = x Z( G ) of x under π satisfies [¯ x, F ( ¯ G )] = [ x Z( G ) , F ( G ) / Z( G )] = [ x, F ( G )] Z( G ) / Z( G ) = ¯1 . Hence ¯ x ∈ C ¯ G ( F ( ¯ G )) = Z( F ( ¯ G )), since C ¯ G ( F ( ¯ G )) ≤ F ( ¯ G ). Therefore x Z( G ) ∈ F ( ¯ G ) = F ( G ) / Z( G ) and x ∈ F ( G ). Since [ x, F ( G )] ⊆ Z( G ) ≤ Z( F ( G )), it nowfollows that x belongs to the second center of F ( G ). Now it is clear that thenilpotency class of M ( G ) is at most 2. This completes the proof of the theorem. (cid:3) We would like to remark here that the first hypothesis in Theorem A, i.e.,C G ( F ( G )) ≤ F ( G ), is naturally satisfied in all finite solvable groups G . So, asa corollary we have Corollary 2.4.
Let G be a finite solvable group such that [ x, F ( G )] is a normalsubset of F ( G ) for every small element x of G . Then M ( G ) is nilpotent of classat most . The following proposition proves the equivalence of the Conjectures A and B stated in the introduction. Proposition 2.5.
Let G be a finite solvable group with trivial center and x be asmall element of G . Then the following are equivalent: (1) x ≤ Z( F ( G )) ; (2) [ x, F ( G )] is a normal subset of F ( G ) .Proof. (1) trivially implies (2). So suppose that (2) holds. Then it follows from theproof of Proposition 2.2 that [ x, F ( G )] ≤ Z( G ) = 1. This shows that x centralizes F ( G ). Since C G ( F ( G )) ≤ F ( G ), it follows that x ≤ Z( F ( G )) and (1) holds. Thiscompletes the proof of the proposition. (cid:3) The following proof of Theorem C is similar to the proof of [1, Theorem A].
Proof of Theorem C.
Write M = M ( G ). Since [ A, x ] is a normal subset of A ,by Proposition 2.2 we get [ A, M ] ≤ Z( G ). Thus [ A, M, M ] = 1. Now by thethree-subgroup lemma, it follows that γ ( M ) ≤ C G ( A ) ≤ A . Thus γ ( M ) =[ γ ( M ) , M, M ] ≤ [ A, M, M ] = 1. This proves that M is nilpotent of class at most3, which completes the proof of the theorem. (cid:3) N SUBGROUPS GENERATED BY SMALL CLASSES IN FINITE GROUPS 5
A group G is said to be of conjugate rank G havethe same conjugacy class size.Finally we prove the following proposition which is not related to the previousdiscussion directly, but is a nice application of Proposition 2.2. Proposition 2.6.
Let G be a finite p -group of conjugate rank . Then G is flat ifand only if the nilpotency class of G is .Proof. Since every group of nilpotency class 2 is flat, we only need to prove theonly if part of the proposition. So let G be flat and x ∈ G − Z( G ). Thus [ x, G ] isa subgroup and hence a normal subgroup of G . Now it follows from the proof ofProposition 2.2 that [ x, G ] ≤ Z( G ). Since γ ( G ) = h [ x, G ] | x ∈ G − Z( G ) i , it followsthat γ ( G ) ≤ Z( G ). This completes the proof of the proposition. (cid:3) Acknowledgements.
I thank Prof. E. C. Dade for his useful comments andsuggestions.
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Subgroups generated by small classes in finite groups , Proc. Amer. Math. Soc. (2008), 2299-2301.[2] K. Ishikawa,
On finite p -groups which have only two conjugacy lengths , Israel J. Math. (2002), 119-123.[3] N. Ito, On finite groups with given conjugate types. I , Nagoya Math. J. (1953). 17-28.[4] A. Mann, Elements of minimal breadth in finite p -groups and Lie algebras , J. Aust. Math.Soc. (2006), 209-214.[5] A. Mann, Correction to ‘Conjugacy class sizes in finite groups’[MR2470541] , J. Aust. Math.Soc. (2009), 429-430.[6] C. H. Sah, Existence of normal complements and extension of characters in finite groups ,Illinois J. Math. (1962), 282-291.[7] C. H. Sah, Automorphisms of finite groups , J. Algebra (1968), 47-68.[8] H. Tandra and W. Moran, Flatness conditions on finite p -groups , Comm. Algebra (2004),2215-2224. School of Mathematics, Harish-Chandra Research Institute, Chhatnag Road, Jhunsi,Allahabad - 211 019, INDIA
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