LOCALLY NILPOTENT LINEAR GROUPS WITH THE WEAK CHAIN CONDITIONS ON SUBGROUPS OF INFINITE CENTRAL DIMENSION

Let V be a vector space over a field F . If G≤GL(V, F ), the central dimension of G is the F -dimension of the vector space V/CV (G). In [DEK] and [KS], soluble linear groups in which the set Licd(G) of all proper infinite central dimensional subgroups of G satisfies the minimal condition and the maximal condition, respectively, have been described. On the other hand, in [MOS], periodic locally radical linear groups in which Licd(G) satisfies one of the weak chain conditions (the weak minimal condition or the weak maximal condition) have been characterized. In this paper, we begin the study of the non-periodic case by describing locally nilpotent linear groups in which Licd(G) satisfies one of the two weak chain conditions. 2000 Mathematics Subject Classification. Primary: 20F22; Secondary: 20H20.


Introduction
Let V be a vector space over a field F .The subgroups of the group GL(V, F ) of all automorphisms of V are called linear groups.The Theory of Linear Groups is one of the most developed branches of the Theory of Groups.If V has finite dimension over F , it is well-known that GL(V, F ) can be identified with the group of non-singular n×n-matrices with entries in F , where n = dim F V .Finite dimensional linear groups have played an important role in mathematics due to the identification mentioned above and the interplay between other mathematical ideas and such groups.However the study of the subgroups of GL(V, F ), when V has infinite dimension over F has received considerably less attention, and requires some additional restrictions.
In the paper [DEK], a way of studying infinite dimensional linear groups that are near to finite dimensional groups, in some sense, was begun.If G is a subgroup of GL(V, F ), then G really acts on the factor-space V /C V (G).Following [DEK], the dimension of this factor-space will be called the central dimension of the subgroup G and will be denoted by centdim F G. Thus, centdim Suppose that G is a linear group of finite central dimension.If C = C G (V /C V (G)), then C is a normal subgroup of G and G/C is isomorphic to some subgroup of GL(n, F ), where n = dim F (V /C V (G)).Since C stabilizes the series C is abelian.Even more, C is torsion-free, if char F = 0, and is p-elementary abelian, if char F = p > 0 (see [KW,Proposition 1.C.3] and [FL,Section 43]).Hence, the structure of G can be determined by the structure of G/C, which is an ordinary finite dimensional linear group.
If G ≤ GL(V, F ), let L icd (G) be the set of all proper subgroups of G of infinite central dimension.In order to study infinite dimensional linear groups G that are close to finite dimensional, it is natural to start making L icd (G) very small in some sense.That is, we want to impose some restriction to L icd (G).Given the success that the study of infinite groups with finiteness conditions has enjoyed, it seemed reasonable to study linear groups with finiteness conditions.Thus, in the paper [DEK] linear groups G in which the set L icd (G) satisfies the minimal condition (or G satisfies Min-icd) have been described.The dual case, that is linear groups G in which L icd (G) satisfies the maximal condition (or G satisfies Max-icd) has been studied in the paper [KS].Since the weak minimal condition and the weak maximal condition are the most natural group-theoretical generalizations of the ordinary minimal and maximal conditions (see [LR,5.1]), it is natural to go on this direction of research with them.
The weak conditions have been introduced by R. Baer [BR] and D. I. Zaitsev [ZD].We recall their definitions in the most general way.Let G be a group and let M be a family of subgroups of G. Then G is said to satisfy the weak minimal condition for M-subgroups, if the set M satisfies the weak minimal condition, that is if given a descending chain of members of M there exists some m ∈ N such that the index |G n : G n+1 | is finite for every n ≥ m.The definition of the weak maximal condition is dual: G is said to satisfy the weak maximal condition for M-subgroups, if the set M satisfies the weak maximal condition, that is if given an ascending chain of members of M there exists some m ∈ N such that the index |G n+1 : G n | is finite for every n ≥ m.In passing we note that groups satisfying one of these weak conditions for different M have been studied by many authors (see [LR,5.1] and the surveys [AK], [KK]).
We say that a group G ≤ GL(V, F ) satisfies the weak minimal condition on subgroups of infinite central dimension (or G satisfies Wmin-icd), if G satisfies the weak minimal condition for L icd (G)-subgroups.Dually, we say that G ≤ GL(V, F ) satisfies the weak maximal condition on subgroups of infinite central dimension (or G satisfies Wmax-icd), if G satisfies the weak maximal condition for L icd (G)-subgroups.Periodic locally radical linear groups satisfying Wmin-icd or Wmax-icd were described in the paper [MOS]: they are either Chernikov or nilpotent-by-abelianby-finite.As a consequence of that description, it was shown that, in the periodic case, the conditions Wmin-icd, Wmax-icd (and even Min-icd) are equivalent.In the current paper, we begin the study of some nonperiodic groups satisfying one of these weak chain conditions as we study locally nilpotent linear groups satisfying Wmin-icd or Wmax-icd.Periodic locally nilpotent linear groups satisfying Wmin-icd or Wmax-icd are Chernikov ( [MOS, Corollary D]), but, as we mentioned above, in the non-periodic case, other types of groups can occur, for example minimax groups.The main results obtained of this paper are the following ones.
Theorem A (Theorem 2.6).Let G be a subgroup of GL(V, F ) of infinite central dimension.Suppose that H is a normal subgroup of G such that G/H is nilpotent.If G satisfies either Wmin-icd or Wmax-icd, then G/H is minimax.In particular, if G is nilpotent, then G is minimax.
From now on we restrict ourselves to the case of char F = p > 0 since the case of char F = 0 requires a different approach.We recall that, if G is a locally nilpotent group, then the set t(G) of all elements having finite order is a characteristic subgroup of G.The subgroup t(G) is called the periodic part or the torsion subgroup of the locally nilpotent group G.
Theorem B (Theorem 3.5).Let G be a locally nilpotent subgroup of GL(V, F ) of infinite central dimension.Suppose that char F = p > 0. If G satisfies either Wmin-icd or Wmax-icd, then G/t(G) is minimax.
Let F be the class of finite groups.If G is a group, then the the intersection G F of all subgroups H of G of finite index is called the finite residual of G.The factor-group G/G F is said to be residually finite.Obviously, G itself is residually finite if G F = 1 .
Theorem C (Theorem 3.6).Let G be a locally nilpotent subgroup of GL(V, F ) of infinite central dimension.Suppose that char F = p > 0. If G satisfies either Wmin-icd or Wmax-icd, then G/G F is minimax and nilpotent.
Let N be the class of nilpotent groups.As above the intersection G N of all normal subgroups H of a group G such that G/H ∈ N is called the nilpotent residual of G and the factor-group G/G N is said to be residually nilpotent.G itself is residually nilpotent if G N = 1 .
Corollary D (Corollary 3.7).Let G be a locally nilpotent subgroup of GL(V, F ) of infinite central dimension.Suppose that char F = p > 0. If G satisfies either Wmin-icd or Wmax-icd, then G/G N is minimax.

Nilpotent linear groups satisfying Wmax-icd or
Wmin-icd In this section we establish that nilpotent linear groups (or more precisely, nilpotent factor-groups of linear groups) that satisfy Wmin-icd or Wmax-icd are minimax or have finite central dimension.In order to show this, we need first some preliminary results, the first lemma of which deals with abelian-by-finite linear groups, and it is widely employed throughout this paper.
If G is a group, as usual, we denote by Π(G) the set of primes occurring as divisors of the orders of the periodic elements of G.
Lemma 2.1.Let G be a subgroup of GL(V, F ) of infinite central dimension.Let H be a normal subgroup of G such that G/H is abelian-by-finite.If G satisfies either Wmin-icd or Wmax-icd, then G/H is minimax.
Proof: Suppose that G has infinite central dimension and G/H is not minimax.Let A be a normal subgroup of G such that A/H is abelian and G/A is finite.By our assumption, A/H is not minimax.We note that if U is some G-invariant subgroup of A with U ≤ H such that A/U is periodic, then [MOS,Lemma 4.4] shows that G/U must be Chernikov.In particular, A/U is likewise Chernikov.
We claim that the 0-rank (or torsioin-free rank ) r 0 (A/H) of A/H is infinite.For, otherwise, we may choose a finite maximal Z-independent subset {a 1 H, . . ., a k H} in A/H.Put B/H = a 1 H, . . ., a k H G/H .Since G/A is finite, B/H is a finitely generated abelian subgroup of A/H such that A/B is periodic.Applying the above paragraph, A/B is a Chernikov group.But in this case A/H is minimax, a contradiction.This contradiction shows our claim, that is r 0 (A/H) is infinite.
Let c 1 H be an element of A/H of infinite order and put There is a positive integer t such that D 1 /H = (C 1 /H) t is free abelian, and, obviously, D 1 /H is also G-invariant.Suppose that we have already constructed an ascending series of G-invariant subgroups of A/H whose factors are free abelian.Then the subgroup D α /H is free abelian (see [KM,Theorem 7.1.3]).
If A/D α is not periodic, then it has an element In particular, it is not Chernikov, which contradicts the fact established above.This final contradiction implies our result.
The proof of a similar theorem for nilpotency is the final step of this section.In order to do this, we need some results about the existence of certain subgroups of a group G provided the first central factor-group of G satisfies prescribed properties.
Lemma 2.3.Let G be a group whose center ζ(G) has an infinite elementary abelian p-subgroup E such that G/E is abelian minimax.Then G has a normal minimax subgroup L such that G = LE.In particular, G/L is an infinite elementary abelian p-group.
Proof: Clearly G is nilpotent.Hence the torsion subgroup T = t(G) of G is characteristic in G. Let P be the Sylow p-subgroup of G. Then E ≤ P .Since G/E is minimax, P/E is a Chernikov subgroup.Let D/E be the divisible part of P/E.Then the abelian group G/D has a finite Sylow p-subgroup P/D and therefore for some subgroup R (see [FL,Theorem 27.5]).In particular, G/R is a finite p-group.
We have where be the upper central series of G/E.We proceed by induction on n.
If n = 1, then G/E is abelian and the statement follows from Lemma 2.3.Suppose that n > 1 and consider the subgroup Z 1 .By Lemma 2.3, Observe that the second center of G includes Z 1 .Therefore for each element zV ∈ Z 1 /V the mapping gV → [gV, zV ] is a homomorphism of G/V into EV /V , whose kernel coincides with C G/V (zV ).Thus [G/V, zV ] is an elementary abelian subgroup.On the other hand, the obvious inclusion shows that the last factor-group is a bounded minimax group, in particular, it is finite.This means that ) is a nilpotent minimax group whose nilpotency class is n − 1. Consequently we can apply now the inductive hypothesis.
Lemma 2.5.Let G be a group and suppose that ζ(G) has a divisible abelian p-subgroup E such that E is not Chernikov and G/E is nilpotent minimax.Then G has a normal minimax subgroup L such that G/L is a divisible abelian p-group which is not Chernikov.
be the upper central series of G/E.Clearly, G is nilpotent.Hence the set T = t(Z 1 ) of all elements of Z 1 having finite order is a characteristic subgroup of Z 1 .Let P be the Sylow p-subgroup of Z 1 .Then E ≤ P .Since G/E is minimax, P/E is a Chernikov subgroup.Let D/E be the divisible part of P/E.Therefore D is periodic nilpotent divisible, so that it is abelian (see [KA,§65]).
Let Q be the Sylow p shows that the latter is a periodic minimax group and hence Chernikov.
Since Z 1 /D is abelian minimax, it has a finitely generated subgroup F/D such that Z 1 /F is periodic.Let F/D = g 1 D, . . ., g t D .
As we have seen above, the subgroups [ Then U is a Chernikov subgroup and In particular; V is normal in G and V /U is a finitely generated abelian group.Furthermore, V is minimax, so that DV /V is not Chernikov.Since the factor-group Z 1 /V is periodic nilpotent, Since DQ 1 /Q 1 is divisible and not Chernikov and Z 1 /DQ 1 is Chernikov, Z 1 /Q 1 has a normal divisible subgroup D 1 /Q 1 of finite index which is not Chernikov.Recall that in a nilpotent periodic group every divisible subgroup lies in the center (see [KW,1.F.1]), so that Z 1 /Q 1 is central-by-finite.As above we see that Clearly L is likewise minimax.It follows that Z 1 /L is a divisible abelian p-group which is not Chernikov.Proceeding in the same way, after finitely many steps we obtain the required result.
Before stating the last result of the section, we recall some definitions and facts that we use subsequently.Let A be an abelian group of finite special rank, M a maximal Z-independent subset of A and set B = M .Put Sp(A) = {p | p is a prime such that the Sylow p-subgroup of A/B is infinite}.
The set Sp(A) is called the spectrum of the group A. If V is also a finitely generated subgroup of A such that A/V is periodic, then are finite, which shows that the set Sp(A) is independent of the choice of the finitely generated subgroup B.
Let G be a nilpotent group of finite special rank and let It is not hard to see that the spectrum of G is the union of the spectrum of all factors of every central series of G.We note that, if H is a normal subgroup of G such that G/H is periodic and p is a prime such that p / ∈ Sp(G), then the Sylow p-subgroup of G/H is finite.
We are now in a position to establish Theorem A, the main result of this section.
be the upper central series of G/H.We proceed by induction on n.
If n = 1, then G/H is abelian and the statement follows from Corollary 2.2.Suppose that n > 1 and we have already proved that G/Z 1 is minimax.We want to prove that Z 1 /H is minimax.Suppose the contrary and seek a contradiction.It suffices to show that G has a normal subgroup U such that G/U is periodic abelian and not minimax.Then Corollary 2.2 will give the required contradiction. Put shows that the latter is periodic Proceeding in the same way, after finitely many steps we construct a normal subgroup E such that L/E is a periodic nilpotent group with Π(L/E) infinite.By [MOS,Corollary 2.7], G has finite central dimension.This contradiction proves that Π(C 1 /B) is finite.
Suppose B has infinite rank.Let p / ∈ Π(C 1 /B).Since B is a free abelian subgroup, B = B p = U .Furthermore B/U is an infinite elementary abelian p-group.It follows that the Sylow p-subgroup of C 1 /U is infinite elementary abelian.Let Q/U be the Sylow p ′ -subgroup of C 1 /U .Then C 1 /Q is an infinite elementary abelian.Applying Corollary 2.4, we see that G has a normal subgroup X such that G/X is an infinite elementary abelian p-group, which contradicts Corollary 2.2.This contradiction shows that Λ is finite and hence B is finitely generated.
Let p ∈ Π(C 1 /B) and let P/B be the Sylow p-subgroup of C 1 /B.As above we can prove that (P/B)/(P/B) p is finite.Then where K/B is divisible and V /B is finite [KL2,Lemma 3].Suppose that K/B is not Chernikov.Then P/V is a divisible p-subgroup which is not Chernikov.By Lemma 2.5, G has a normal subgroup W such that G/W is a divisible p-subgroup which is not Chernikov, which contradicts Corollary 2.2.This contradiction shows that K/B and P/B are Chernikov subgroups.This and the finiteness of Π(C 1 /B) give that C 1 /B is Chernikov.Since B is a finitely generated abelian subgroup, this means that C 1 is minimax.Thus Z 1 /H is minimax and consequently G/H is likewise minimax.
Corollary 2.7.Let G be a nilpotent subgroup of GL(V, F ) of infinite central dimension.If G satisfies either Wmin-icd or Wmax-icd, then G is minimax.

Locally nilpotent linear groups with Wmin-icd or
Wmax-icd In the sequel, we usually restrict ourselves to the case of char F = p > 0. Other different techniques are needed to develop the case of characteristic 0. The first result is interesting in its own right and will be very useful because it gives the structure of finite central dimensional subgroups of an infinite central dimensional locally nilpotent linear group satisfying Wmin-icd or Wmax-icd.We remark that, in prime characteristic, finite central dimensional subgroups of these groups are soluble (see [MOS,Lemma 4.1] or at the beginning of Section 4 of [DEK]).
Proposition 3.1.Let G be a locally nilpotent subgroup of GL(V, F ) of infinite central dimension, where char F = p > 0. Suppose that G satisfies either Wmin-icd or Wmax-icd.If H is a normal subgroup of G of finite central dimension and P is the Sylow p-subgroup of H, then (1) P is nilpotent and bounded, (2) H/P is minimax and abelian-by-finite.In particular, H/t(H) is an abelian minimax group and the Sylow p ′ -subgroup of H is Chernikov.
Since centdim F H is finite, C has finite codimension in V .Therefore the series C ≤ V can be refined to a composition F G-series.That is, V has a finite series of F G-submodules Each factor-group G/D j+1 can be embedded into GL(C j+1 /C j , F ) and, since C j+1 /C j has finite dimension over F , G/D j+1 is soluble (see [WB,Corollary 3.8]).By a result due to Malcev (see [WB,Lemma 3.5]), G/D j is abelian-by-finite and then Lemma 2.1 yields that G/D j+1 is minimax.that locally nilpotent torsion-free linear groups with Wmin-icd or Wmaxicd are hyperabelian.
Lemma 3.3.Let G be a locally nilpotent group.Suppose that G has an ascending series of normal subgroups are all non-trivial torsion-free abelian minimax groups.Then C has a G-invariant subgroup B such that C/B is periodic and Π(C/B) is infinite.
Proof: Proceeding by induction on n, we are going to construct an ascending series of G-invariant subgroups Since C 1 is nilpotent minimax, then we can choose a prime q such that C q 1 = C 1 .Put p 1 = q and B 1 = C q 1 .It easy to see that B 1 satisfies the above conditions.Suppose that we have already found primes p 1 , . . ., p k and constructed G-invariant subgroups Hence the claimed construction have just been carried out.
Put now B = ∪ n∈N B n .From (c) we have Lemma 3.4.Let G be a locally nilpotent subgroup of GL(V, F ). Suppose that G satisfies either Wmin-icd or Wmax-icd.Then G/t(G) has an ascending series of normal subgroups whose factors are pure abelian.In particular, G/t(G) is hyperabelian.
Proof: Put T = t(G).If G has finite central dimension, then by [MOS,Lemma 4.1] G is soluble.Therefore we can assume that centdim F G is infinite.Since G/T is a torsion-free locally nilpotent group, G/T has a central series Z with pure subgroups ([GV, Theorem 7]).In particular, G/T has a non-identity normal pure subgroup U/T .If U has finite central dimension, then again by [MOS,Lemma 4.1] U is soluble.Therefore U/T is soluble and so U/T has a non-identity characteristic abelian subgroup A/T .Hence A/T is a non-identity abelian normal subgroup of G/T .Put L/A = t(G/A).Then L/T is a pure abelian normal subgroup of G/T (see [KA,§66 and §67]).Suppose now that centdim F U is infinite for every normal pure subgroup U/T of G/T .We remark that if U/T and W/T are two pure normal subgroups of G/T with the property U/T < W/T , then the factor W/U has to be infinite.It is rather easy to see that if G satisfies Wmin-icd (respectively Wmax-icd), then G/T satisfies the minimal condition for pure normal subgroups (respectively the maximal condition for pure normal subgroups).
Let Z be a central series of G.If G satisfies Wmin-icd, then a set of subgroups of Z linearly ordered by inclusion satisfies the minimal condition.Then the set of subgroups of Z is completely ordered by inclusion.In particular, Z has a minimal element B/T .Since Z is a central series, B/T ≤ ζ(G/T ).In particular, G/T has a pure abelian normal subgroup.
If G satisfies Wmax-icd, then G/T is a nilpotent group of finite special rank by [GV,Theorem 10].Therefore, in this case G/T has a pure abelian normal subgroup too.It suffices to apply transfinite induction to obtain the required result.
After these two lemmas, we are now in a position to show the remainder of the main results of this paper.
Theorem 3.5.Let G be a locally nilpotent subgroup of GL(V, F ) of infinite central dimension.Suppose that char F = p > 0. If G satisfies either Wmin-icd or Wmax-icd, then G/t(G) is minimax.
Proof: Put T = t(G).If G/T is soluble, the result follows from Corollary 3.2.Therefore, we suppose that G is insoluble.By Lemma 3.4, G/T is hyperabelian.Therefore we can construct an ascending series of normal subgroups such that C α+1 /C α is a maximal abelian normal subgroup of G/C α , for every ordinal α < γ.Proceeding just as in the proof of Lemma 3.4, we see that every subgroup C α /T is pure in G/T .Since G is insoluble, γ ≥ ω, where ω is the first infinite ordinal.Put C = C ω .Every subgroup C n is soluble for each n ∈ N. Therefore C n /T is minimax for each n ∈ N by Corollary 3.2.Lemma 3.3 shows that C has a G-invariant subgroup B such that C/B is periodic and Π(C/B) is infinite.Since G is locally nilpotent, C/B is a direct product of infinitely many Sylow subgroups.By [MOS,Corollary 2.7], centdim F C is finite, and, by [MOS,Lemma 4.1], C must be soluble, a contradiction.This contradiction shows that G/T is soluble, as required.
Theorem 3.6.Let G be a locally nilpotent subgroup of GL(V, F ) of infinite central dimension.Suppose that char F = p > 0. If G satisfies either Wmin-icd or Wmax-icd, then G/G F is minimax and nilpotent.
Proof: Let M be the family of all normal subgroups H such that G/H is finite, and let L = G F be the finite residual of G, so that L = {M | M ∈ M}.Let q be a prime and put M(q) = {H ∈ M | G/H is a finite q-group}.In other words, QL/L is isomorphic to some subgroup of G/G F .By Theorem 3.6, G/G F is minimax.Thus QL/L is finite.Since (G/L)/(QL/L) is minimax, G/L is likewise minimax.
abelian and Π((L/D)/C L/D (cD)) ⊆ Π(C 1 /D).In particular, Π((L/D)/C L/D (cD)) ∩ Sp(L/C 1 ) = ∅.On the other hand, the obvious inclusion C 1 /D ≤ C L/D (cD) implies that (L/D)/ C L/D (cD) is a nilpotent minimax group and Sp((L/D)/C L/D (cD)) ⊆ Sp(L/C 1 ).Thus the Sylow q-subgroup of (L/D)/C L/D (cD) is finite for all q ∈ Π((L/D)/C L/D (cD)).In particular, the set Π((L/D)/C L/D (cD)) is finite and hence (L/D)/C L/D (cD) is likewise finite.This means that C 2 /D ≤ F C(L/D) where F C(L/D) is the F C-center of L/D.The factor-group C 2 /C 1 has finite rank, therefore C 2 includes a finite set of elements c 1 and an infinite set of distinct primes {p n | n ∈ N} satisfying the following conditions: (a) B n ≤ C n , (b) C n /B n is finite and Π(C n /B n ) = {p 1 , . . ., p n }, where p 1 , p 2 , . . ., p n are distinct primes and (c) B n ∩ C n−1 = B n−1 , for all n ∈ N.
satisfying conditions (a)-(b)-(c).We will find a new prime p k+1 and construct a subgroup B k+1 in such a way (a)-(b)-(c) hold.C k+1 /B k is an extension of a finite subgroup C k /B k by a torsion-free abelian group C k+1 /C k .Being locally nilpotent, C k+1 /B k is nilpotent.It is easy to see that t(C k+1 /B k ) = C k /B k .On the other hand, by [HK, Proposition 2], C k+1 /B k has a torsion-free normal subgroup D k+1 /B k show that (QL/L)/((H/L) ∩ (QL/L)) is finite.Put U/L = (H/L) ∩ (QL/L).We haveT /L = QL/L × R/L, where R/L is the Sylow p ′ -subgroup of T /L.Put W/L = (U/L)(R/L).It follows that (T /L)/(W/L) is a finite p-group.Since G/T is nilpotent and T /W is finite, G/Wis nilpotent too.Then it has a torsion-free normal subgroup D/W such that G/D is bounded ([HK, Proposition 2]).Since G/W has finite special rank, G/D, being bounded, is finite.Furthermore, D/W is torsion-free, so (T /L) ∩ (D/L) = W/L.The equation L = M shows that for each element xL ∈ QL/L there is a normal subgroup W/L of finite index such that xL / ∈ W/L.It follows that (QL/L) ∩ (G F /L) = 1 .
Corollary 2.4.Let G be a group whose center ζ(G) has an infinite elementary abelian p-subgroup E such that G/E is nilpotent minimax.Then G has a normal subgroup L such that G/L is an infinite elementary abelian p-group.
[HK, Proposition 2], R/Q has a normal torsion-free subgroup W/Q such that R/W is a bounded p-group.Proceeding as in the proof of[HK, Proposition 2], we see that the subgroup W/Q can be chosen such that W/Q is characteristic in R/Q, and hence W is normal in G. Since W/Q is torsion-free, W is minimax.The finiteness of G/R implies that G/W is bounded.Hence G/W is a central extension of EW/W by a bounded minimax group G/EW .However a bounded abelian minimax group is finite.In other words, G/W is central-byfinite.It follows that G/W has a normal finite subgroup L/W such that G/W = (L/W )(EW/W ).Clearly L is likewise minimax.It follows that G/L is an infinite elementary abelian p-group, as required.