aa r X i v : . [ m a t h . R A ] J a n GROUP ALGEBRA WHOSE UNIT GROUPIS LOCALLY NILPOTENT
V. BOVDI
Dedicated to the memory of Professor L.G. Kov´acs
Abstract.
We present a complete list of groups G and fields F for which: (i) the group ofnormalized units V ( F G ) of the group algebra
F G is locally nilpotent; (ii) the group algebra
F G has a finite number of nilpotent elements and V ( F G ) is an Engel group. Introduction
Let V ( F G ) be the normalized subgroup of the group of units U ( F G ) of the group algebra
F G of a group G over a field F of characteristic char( F ) = p ≥
0. It is well known that U ( F G ) = V ( F G ) × U ( F ), where U ( F ) = F \ { } . The group of normalized units V ( F G ) of amodular group algebra
F G has a complicated structure and was studied in several papers. Foran overview we recommend the survey paper [2].An explicit list of groups G and rings K for which V ( KG ) are nilpotent was obtained byI. Khripta (see [16] for the modular case and [17] for the non-modular case). In [3] it wascompletely determined when V ( F G ) is solvable. It is still a challenging problem whether V ( F G ) is an Engel group. This question has a long history (see [4, 5, 6, 9, 10, 24]). Thenon-modular case was solved by A. Bovdi (see [4], Theorem 1.1, p.174). For the modular casethere is no complete solution (see [4], Theorem 3.2, p.175), but there is a full description of
F G when V ( F G ) is a bounded Engel group (see [4], Theorem 3.3, p.176).It is well known (for example, see [26]) that the Engel property of a group is close to itslocal nilpotency. A locally nilpotent group is always Engel (see [26]). However these classesof groups do not coincide (see the Golod’s counterexample in [11]). The following results areclassical (see [26]): each Engel profinite group (see [27]), each compact Engel group (see [18]),each Engel linear group (see [25]), each 3-Engel and 4-Engel group (see [15] and [14]) and allEngel groups satisfying max (see [1]) are locally nilpotent.A group G is said to be Engel if for any x, y ∈ G the equation ( x, y, y, . . . , y ) = 1 holds,where y is repeated in the commutator sufficiently many times depending on x and y . We shalluse the left-normed simple commutator notation ( x , x ) = x − x − x x and( x , . . . , x n ) = (cid:0) ( x , . . . , x n − ) , x n (cid:1) , ( x , . . . , x n ∈ G ) . A group is called locally nilpotent if all its f. g. (finitely generated) subgroups are nilpotent.Such a group is always Engel (see [26]). The set of elements of finite orders of a group G (whichis not necessarily a subgroup) is called the torsion part of G and is denoted by t ( G ). We usethe notion and results from the book [8] and the survey papers [2, 26]. Mathematics Subject Classification.
Key words and phrases. unit group, group algebra, locally nilpotent group, Engel group.Supported by the UAEU grants: UPAR G00001922.
In several articles, M. Ramezan-Nassab attempted to describe the structure of groups G forwhich V ( F G ) are Engel (locally nilpotent) groups in the case when
F G have only a finitenumber of nilpotent elements (see Theorem 1.5 in [21], Theorems 1.2 and 1.3 in [20] andTheorem 1.3 in [22]). The following theorem gives a complete answer.
Theorem 1.
Let
F G be the group algebra of a group G . If F G has only a finite number ofnon-zero nilpotent elements, then F is a finite field of char( F ) = p . Additionally, if V ( F G ) isan Engel group, then V ( F G ) is nilpotent, G is a finite group such that G = Syl p ( G ) × A , where Syl p ( G ) = h i , G ′ ≤ Syl p ( G ) and A is a central subgroup of G . The next result plays a technical role.
Theorem 2.
Let G be a group such that G ′ is a locally finite p-group and the p -Sylow subgroup P = Syl q ( G ) of G is normal in G . If V ( F G ) is an Engel group and char( F ) = p > then thefollowing conditions hold: (i) V ′ ≤ I ( G ′ ) and I ( P ) = Syl p ( V ( F G )) ; (ii) for each q = p , the group Syl q ( G ) is central; (iii) the group t ( G ) = P × D , where D = × q = p Syl q ( G ) ; (iv) let M be the subgroup of G/P generated by the central subgroup
DP/P and by all the ele-ments of infinite order in
G/P . Then V ( F G ) / (1+ I ( P )) ∼ = V ( F M ) and V ( F M ) /V ( F D ) is a torsion-free group; (v) G/ t ( G ) is an abelian torsion-free group and, if D is finite, then V ( F M ) ∼ = V ( F D ) × ( G/ t ( G ) × · · · × G/ t ( G ) | {z } n ) , (1) where n is the number of summands in the decomposition of F D into a direct sum offields.
The next two theorems completely describe groups G with V ( F G ) locally nilpotent. Somespecial cases of Theorem 3 were proved by I. Khripta (see [17]) and M. Ramezan-Nassab (seeTheorem 1.2 in [20] and Corollary 1.3 and Theorem 1.4 in [21]).
Theorem 3.
Let
F G be a modular group algebra of a group G over the field F of positivecharacteristic p . The group V ( F G ) is locally nilpotent if and only if G is locally nilpotent and G ′ is a p -group. Theorem 4.
Let
F G be a non-modular group algebra of characteristic p ≥ . The group V ( F G ) is locally nilpotent if and only if G is a locally nilpotent group, t ( G ) is an abelian groupand one of the following conditions holds: (i) t ( G ) is a central subgroup; (ii) F is a prime field of characteristic p = 2 t − , the exponent of t ( G ) divides p − and g − ag = a p for all a ∈ t ( G ) and g ∈ G \ C G ( t ( G )) . As a consequence of Theorems 2 and 3 we obtain the classical result of I. Khripta.
Corollary 1.
Let
F G be a modular group algebra of positive characteristic p . The group V ( F G ) is nilpotent if and only if G is nilpotent and G ′ is a finite p -group.The structure of V ( F G ) is the following: the group t ( G ) = P × D , where P is the p -Sylow subgroup of G , D is a central subgroup, I ( P ) is the p -Sylow subgroup of V ( F G ) and OCALLY NILPOTENT UNIT GROUP 3 V ′ ≤ I ( G ′ ) . Moreover, if D is a finite abelian group, then we have the following isomorphismbetween abelian groups V ( F G ) / (cid:0) I ( P ) (cid:1) ∼ = V ( F D ) × ( G/ t ( G ) × · · · × G/ t ( G ) | {z } n ) , where n is the number of summands in the decomposition of F D into a direct sum of fields. Proofs
We start our proof with the following important
Lemma 1.
Let
F G be the group algebra of a group G such that V ( F G ) is an Engel group. If H is a subgroup of t ( G ) and H does not contain an element of order char( F ) , then H is abelian,each subgroup of H is normal in G and each idempotent of F H is central in
F G .Proof. If a ∈ t ( G ) and g ∈ G \ N G ( h a i ), then x = b ag ( a − = 0 and 1 + x ∈ V ( F G ). Astraightforward calculation shows that(1 + x, a, m ) = 1 + x ( a − m , ( m ≥ . Since V ( F G ) is an Engel group, x ( a − s = 0 and x ( a − s +1 = 0 for a suitable s ∈ N . Itfollows that x ( a − s a i = x ( a − s for each i ≥ x ( a − s · | a | = x ( a − s (1 + a + a + · · · + a | a |− ) = 0 . Since char( F ) does not divide | a | , we have that x ( a − s = 0, a contradiction. Hence everyfinite cyclic subgroup of H is normal in G , so H is either abelian or hamiltonian.If H is a hamiltonian group, then using the same proof as in the second part of Lemma 1.1of [10] (see p.122), we obtain a contradiction.Hence H is abelian and each of its subgroups is normal in G .We claim that all idempotents of LT are central in L h T, g i , where T is a finite abeliansubgroup of H , g ∈ G and L is a prime subfield of the field F .Assume on the contrary that there exist a primitive idempotent e ∈ LT and g ∈ G such that geg − = e . The element b = g − a − ga = 1 for some a ∈ T and W = h c − ( g − cg ) | c ∈ T i = h T, g i ′ ⊳ h T, g i is a non-trivial finite abelian subgroup of T .Obviously each subgroup of T ≤ H is normal in h T, g i , so the idempotent f = | W | d h W i ∈ L [ T ]is central in L h T, g i and can be expressed as f = f + · · · + f s in which f , . . . , f s are primitiveand mutually orthogonal idempotents of the finite dimensional semisimple algebra L [ T ].The idempotent e does not appear in the decomposition of f . Indeed, otherwise we have ef = e . If e = P i α i t i , where α i ∈ L and t i ∈ T , then g − ( ef ) g = g − egf = X i α i g − t i gf = X i α i t i ( t i , g ) f = ef so e = f e is central, a contradiction.If ef = 0, then ef = e , again a contradiction. Thus ef = 0.Consider f ∗ = | b | c h b i ∈ L [ T ]. Since h b i ⊆ W , the idempotent f ∗ appears in the decompositionof f , so ef ∗ = 0. Furthermore geg − ∈ L [ T ] is primitive as an automorphism’s image of the V. BOVDI primitive idempotent e , so ege = e ( geg − ) g = 0. Evidently (1 + eg ) − = 1 − eg and(1 + eg, a ) = (1 − eg )(1 + ea − ga )= (1 − eg )(1 + egb ) = 1 + eg ( b − . Now an easy induction on n ≥ eg, a, n ) = 1 + eg ( b − n . However V ( F G ) isEngel, so there exists m such that eg ( b − m = 0 and eg ( b − m +1 = 0 . Thus eg ( b − m b i = eg ( b − m for any i ≥
0, which leads to the contradiction eg ( b − m = eg ( b − m f ∗ = eg ( b − m − (cid:0) ( b − f ∗ (cid:1) = 0 . Therefore, all idempotents of
F H are central in
F G . (cid:3) Proof of Theorem 1.
Let N ∗ be the (finite) set of all nilpotent elements of F G \
0. If x ∈ N ∗ then the subset of nilpotent elements { λx | λ ∈ F } is finite, so F is a finite field of char( F ) = p .Assume that there exist u ∈ N ∗ and g ∈ G \ t ( G ) such that gu = ug . Since the rightannihilator L of the left ideal I l ( h g i ) = h g i − | i ∈ Z i F G is different from zero if and only if h g i is a finite group (see Proposition 2.7, [8], p.9), the set { u ( g i − | i ∈ Z } is an infinite subsetof N ∗ , a contradiction.Let 0 = u ∈ N ∗ . Clearly g − i ug i = 0 and g − i ug i ∈ N ∗ for any g ∈ G \ t ( G ) and i ∈ Z . Since N ∗ is a finite set, at least for one i ∈ Z there exists j ∈ Z ( j = i ) such that g − i ug i = g − j ug j ,so g i − j u = ug i − j , a contradiction. Consequently G = t ( G ), i.e. G is a torsion group.If Syl p ( G ) = h i , then each subgroup of t ( G )(= G ) is abelian and normal in G by Lemma 1.Thus F G is a direct sum of fields, so N ∗ = ∅ , which is impossible.Let P = Syl p ( G ) = h i . Evidently { h − | h ∈ P } is a finite subset of nilpotent elements,so P is a finite subgroup in G . Assume that P is not normal in G . Since P < V ( F G ) is Engeland nilpotent, some element of P must have a conjugate w ∈ N G ( P ) \ P (see [19], LemmaV.4.1.1, p.379 [p.307 in the English translation]). Consequently P (cid:8) h P, w i and h P, w i is afinite p -group, a contradiction.Hence P is a normal subgroup in G and the subset of nilpotent elements { ( h − gλ | h ∈ P, g ∈ G, λ ∈ F } is finite, so G is finite as well. Since every finite Engel group is nilpotent by Zorn’s theorem (see[23], 12.3.4, p.372), G is a finite nilpotent group which is a direct product of its Sylow subgroups(see [12], 10.3.4, p.176) and G ′ is a finite p -group (see [4], Theorem 3.2, p.175). Moreover any q -Sylow subgroup ( q = p ) is an abelian normal subgroup in G and each of its subgroups is alsonormal in G by Lemma 1. Consequently the finite group G = P × A , where A = ∪ q = p Syl q ( G )is an abelian normal subgroup of G . Moreover, if g ∈ G and z ∈ A , then( z, g ) = z − ( g − zg ) ∈ P ∩ A = h i so A is central. Since F G is a finite algebra, V ( F G ) is nilpotent (see [23], 12.3.4, p.372). (cid:3)
Proof of Theorem 2.
Clearly G ′ is locally nilpotent and the ideal I ( P ) is locally nilpotent, hence1 + I ( P ) is a locally nilpotent p -group which coincides with the p -Sylow normal subgroup of V ( F G ). In view of G ′ ⊆ P we obtain that V ( F G ) / (cid:0) I ( P ) (cid:1) ∼ = V ( F [ G/P ]) , V ( F G ) / (cid:0) I ( G ′ ) (cid:1) ∼ = V ( F [ G/G ′ ]) (2) OCALLY NILPOTENT UNIT GROUP 5 are abelian groups. From the second isomorphism in (2) it follows that V ′ ≤ I ( G ′ ).Let Syl q ( G ) be a Sylow q -subgroup ( q = p ) of G . According to Lemma 1, Syl q ( G ) is a normalabelian subgroup in G . Moreover, if g ∈ G and z ∈ Syl q ( G ), then( z, g ) = z − ( g − zg ) ∈ P ∩ Syl q ( G ) = h i so Syl q ( G ) is central. Consequently, the group t ( G ) is equal to P × D where D = × q = p Syl q ( G ) ⊆ ζ ( G ) and ζ ( G ) is the center of G .Define the subgroup M of G/P to be the group generated by the central subgroup
DP/P and by all elements of infinite order in
G/P . Evidently, each element of M can be expressedas a product of an element from DP/P ( ∼ = D ) and an element of infinite order. Consequently,the subgroup M of G/P has no element of order p and M/D is a free abelian group.The algebra F [ M ] can be expressed as a crossed product S ( K, F [ D ]) of the group K ∼ = M/D over the group algebra F [ D ] (see [8], Lemma 11.1, p.63). Since F [ D ] isomorphic to a centralsubalgebra of S ( K, F [ D ]), the algebra S ( K, F [ D ]) is a twisted group ring with G -basis { t g | g ∈ G } such that t g t h = t gh µ g,h for all g, h ∈ G , where µ g,h ∈ µ = { µ a,b ∈ U ( F D ) | a, b ∈ G } is the factor system of S ( K, F D ) (see [8], Chapter 11). Any unit u ∈ F M can be written as u = P si =1 α i t g i ∈ S ( K, F D ), where α i ∈ F D . Define B = h supp ( α ) , . . . , supp ( α s ) i ≤ D. If e , . . . , e n is a complete set of primitive, mutually orthogonal idempotents of the f.d. com-mutative algebra F B such that e + · · · + e n = 1, then F B = F Be ⊕ · · · ⊕ F Be n and F M = F M e ⊕ · · · ⊕ F M e n . The field
F Be i is denoted by F i . It is easy to see that u ∈ S ( K, F B ) = S ( K, F ⊕ · · · ⊕ F n ) = S ( K, F ) ⊕ · · · ⊕ S ( K, F n ) . Since K is an ordered group, all units in S ( K, F i ) are trivial (see [7], Lemma 3, p.495) and ue i = β i e i t h i for some h i ∈ K and β i ∈ F B . Consequently u = ( β e ) t h + ( β e ) t h + · · · + ( α n e n ) t h n , ( β i ∈ F B, h i ∈ K )so U (cid:0) S ( K, F i ) (cid:1) ∼ = U ( F i ) × K . Note that the description of the group V ( F M ) depends only onthe subgroup D .Let us give all invariants of V ( F M ) in the case when the abelian group D is finite. Set B = D . Since U ( S ( K, F i )) ∼ = U ( F i ) × K for i = 1 , . . . , n , U ( F [ M ]) ∼ = U ( S ( K, F )) × · · · × U ( S ( K, F n )) ∼ = ( U ( F ) × K ) × · · · × ( U ( F n ) × K ) ∼ = ( U ( F ) × · · · × U ( F n )) × L, where L is a direct product of n copies of K . It follows that U ( F M ) is an extension of thetorsion-free abelian group L by a central subgroup U ( F D ). (cid:3) Proof of Theorem 3.
Since V ( F G ) is a locally nilpotent group, G is also locally nilpotent. Let S = h f , . . . , f s | f i ∈ V ( F G ) i be an f. g. subgroup of V ( F G ). Clearly H = h supp ( f ) , . . . , supp ( f s ) i is an f. g. nilpotent subgroup of G and S ≤ V ( F H ) < V ( F G ). Hence we may restrict ourattention to the subgroup V ( F H ), where H is an f. g. nilpotent subgroup of G . V. BOVDI
Let H be an f. g. nilpotent group containing a p -element and let g, h ∈ H such that ( g, h ) = 1.Obviously t ( H ) is finite (see [13], 7.7, p. 29) and the finite subgroup Syl p ( t ( H )) is a directfactor of t ( H ) (see [12], 10.3.4, p.176). Consequently, there exists c ∈ ζ ( H ) of order p , and b c = P p − i =0 c i is a square-zero central element of F G . Since L = h g, h, g b c i is an f. g. nilpotentsubgroup of V ( F G ), so L is Engel and there exists m ∈ N such that the nilpotency class cl ( L )of L is at most p m .Let us show that ( g, h ) is a p -element. Indeed, if q = p m then we have1 = (cid:0) g b c, h, q (cid:1) = 1 + b c q X i =0 ( − i (cid:0) qi (cid:1) g h q − i = 1 + b c ( g h q − g ) , because (cid:0) qi (cid:1) ≡ p ) for 0 < i < q . It follows that b c ( g h q − g ) = 0 and ( g, h q ) ∈ supp ( b c ),which yields that g h q = c i g for some 0 ≤ i < p . Hence (cid:0) h − q · · · h − q | {z } p ) g ( h q · · · h q | {z } p (cid:1) = ( c i ) p g = g and h p m +1 ∈ C G ( g ), so h p m +1 ∈ ζ ( L ) for all h ∈ L and L ′ is a finite p -group by a theorem ofSchur (see [23], 10.1.4, p.287). Consequently G ′ is a p -group.Conversely, it is sufficient to prove that if H is an f. g. nilpotent subgroup of G such that H ′ is a p -group then V ( F H ) is nilpotent. We known that all subgroups and factor groups of H are also f. g. groups ([23], 5.2.17, p.132). Moreover t ( H ) = P × D is finite (see [23], 12.1.1,p.356) and H/ t ( H ) is a direct product of a finite number of infinite cyclic groups.Let Syl q ( H ) be a Sylow q -subgroup ( q = p ) of H . It is well known (see [13], 7.7, p. 29 and[12], 10.3.4, p.176) that Syl q ( H ) is normal in H . Moreover, if g ∈ H and z ∈ Syl q ( H ), then( z, g ) = z − ( g − zg ) ∈ P ∩ Syl q ( G ) = h i so Syl q ( H ) is central. Consequently, t ( H ) = P × D , where D = × q = p Syl q ( G ) ⊆ ζ ( H ) and (1)holds by Theorem 4(v). Hence V ( F [ G/P ]) is an extension of an f. g. abelian group L by acentral subgroup V ( F D ). By Lemma 2.4 from [6] (or by Lemma 2.1 in [4], on p.176) everyextension of a nilpotent group 1 + I ( P ) by an f. g. abelian group L is nilpotent. Since U ( F D )is central so the group U ( F H ) is nilpotent. (cid:3)
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