The Nilpotency Class of Fitting Subgroups of Groups with Basis Property

International Journal of Algebra, Vol. 6, 2012, no. 14, 697 - 704

M. Alkadhi, A. Al Khalaf and M. Quick

Al-Imam University, KSA St-Andrews University Scotland, UK [email protected] [email protected] [email protected]

Abstract

A group G is called group with basis property if for any subgroup H of G there exists a basis, i.e. a minimal generating set such that any two bases of H are equivalent. We show that any group with basis property is Frobenies group with kernel p-group P = Fit(G) and complement q-group of order qb. Let G be group with basis property, then Fit(G) in general has no upper bound, but in some special cases the nilpotency class of Fit(G)is bounded and that depends on the order of the complement group. We proved the following theorem: Let G be a ﬁnite group with basis property which is not a p−group, then the nilpotency class of the Fitting subgroup Fit(G) has no upper bound.Moreover, for each prime number q>2there exists a nite group with the basis property such that: c(Fit(G)=q − 1; | | = q.

Keywords: group with basis property, nilpotency class, Fitting subgroup, Frobenius group

1 Introduction

It is well-known that for a nitely generated vector space, all its minimal generating sets have same cardinality. It is a rather rare phenomenon in other algebraic structures, though, to have a dimension. For example consider the abelian groupZ6 it has both {1} and {2, 3} as minimal generating sets. 698 M. Alkadhi, A. Al Khalaf and M. Quick

2 Preliminaries

Deﬁnition 2.1 An inverse semigroup (in particular a group) S is called an inverse semigroup with basis property, if each two minimal (irreducible) generating sets (with respect to inclusion) of an arbitrary subsemigroup H of S are equivalent i.e., they have the same cardinality [7].

Example 2.2 vector space, free module over principal ring, free commutative semigroup, and free inverse semigroup.

In[8] some properties of groups with basis property were stated among them the followings:

1. Any group with basis property is a periodic group, i.e. each element has ﬁnite order.

2. Any group with basis property is solvable.

3. Any ﬁnite p-group P is a group with basis property.

4. A Homomorphic image of any ﬁnite group with basis property is also a group with basis property.

5. Every abelian p-group P has basis property.

In [1] we have showed that any group with basis property is quasiprimary, meaning that the order of any of its elements is either a power of prime number p or a power of another prime number p = q. Even though every finite p-group is satisfies basis property, the reverse is not true in general. In [4] it was shown that every finite quasiprimary non-primary solvable group has order paqb; p = q, a, b ∈ N and in this case it is an extension p-group P = Fit(G)by a group H,where H is one of the following types:

1. H is a cyclic q-group with order qb;

2. H is a generalized quaternion group;

3. H is a polycyclic (metacyclic) group.

Moreover any automorphism on P induced by q-element from G \ H is regular.Thus the class of quasiprimary solvable group- including groups with basis property - has been determined to have an obvious form. In [2], and [3]we showed that the last two cases above do not satisfy basis property, which mean that any finite group with basis property which is not p-group, must be a semidirect product of p-groupP by a cyclic q-group of orderqb. While The nilpotency class of fitting subgroups 699 in the first case above, where a group G is a semidirect product of p-group P by cyclic q-group of order qb in [4], we described of the automorphism ϕ on P induced by . The automorphism ϕ defines a linear representation of group on every quotient Frattini subgroup of group P .

• Frattini subgroups : Φ0(P )=P, Φ1(P )=Φ(P ),...,Φj+1(P ) = Φ(Φj (P ))

1 2 • Quotient Frattini subgroups of a group P : P0 = P/Φ(P ),P1 =Φ(P )/Φ (P ),...,Pj = j j+1 Φ (P )/Φ (P ), where j =0,...,jp and jp = max{j ||Pj| > 1}

We can consider the quotient Frattini subgroup as vector space over the ﬁeld GF (p). In [1] we proved the following theorem:

Theorem 2.3 Let a ﬁnite group G be a semidirect product of a p-group P = Fit(G) (Fitting subgroup) of G by a cyclic q−group of order qb, where p = q (p and q are prime numbers), b ∈ N. Then the group G has basis property, if and only if, for any element u ∈, u = e and for any invariant subgroup H of P , the automorphism ϕu must deﬁnes an isotopic representation on every quotient Frattini subgroup H.

Lemma 2.4 Let G be a group with basis property which is not p-group. Then G is Frobenius group with kernel p-group P and its complenent group is cyclic group with order qb, where p = q are prime numbers. Proof: Let G be a group with basis property and non p-group. Then by [1] G is semidirect product of a p-group P by a cyclic q−group of orderqb, since P G. Thus P is p−Sylow subgroup of G . From[16], P containes all p-elements. Since G is group with basis property, then by [1] it is a quasiprimary group, where a p-element does not commute with a q-element. So for any e = x and x ∈ P , CG(x) ≤ P , meaning that P is a complete isolator according to [12]. Since P G, then by [13], P is a kernel of the Frobenius group G, moreover is a complement of G . Hence if we take H any complement of G such that G is a semidirect product of a p-group P by H and G| = |P ||H| = |P |qb, i.e. H is q- Sylow subgroup of G, then H must be a cyclic group of order qb and a conjugate of . Thus is a complement of G.

We saw before that any group with basis property is a Frobenius group with kernel a Fitting subgroup Fit(G)=P and its complement is a cyclic group.So we can use known results for the nilpotency class of the kernel in a Frobenius group to describe the nilpotency class of the Fitting subgroup of the group with basis property. As was shown in[12],[14], [15], in general the nilpotency class of Fitting subgroups of a Frobenius group has no upper bound and it depends on the order of the complement of the Frobenius group. 700 M. Alkadhi, A. Al Khalaf and M. Quick

Lemma 2.5 Let G = Pλ < y > be a Frobenius group such that | | = qb; |P | = pc, p = q are primes and b, c ∈ N.Then the nilpotency class c(P ) of the Fitting subgroup P equals at most k(q) (So k(q) is an upper bound) and satisﬁes the following:

1. If q =2, then P is an abelian group, i.e k(2) = 1.

2. If q =3, then P is a metaabelian group, i.e k(3) = 2.

3. If q =5, then k(5) = 6.

4. If q =7, then k(7) = 12.

q−1 q2−1 ≤ ≤ (q−1)2 −1 5. If q is odd number, then 4 k(q) q−2 .

6. If pc = εqb +1 such that ε ∈{1, 2}, b, c ∈ N, then P is an abelian group.

7. If P is not an abelian group, then pc ≥ 27.

Proof. Let G = Pλ < y > be a Frobenius group, then:

1. In [16] showed that if a regular automorphism of order 2 can be deﬁned on a p-group P , then P must be abelian, resulting that k(2) = 1.

2. In [12] it is proved that if q =3, then P is a metaabelian group, resulting that k(3) = 2.

3. In [18] it was shown that if q =5, then the nilpotency class of P is 6, resulting that k(5) = 6.

4. In [6] it shown that if q =7, then the nilpotency class of P is 12, resulting k(7) = 12.

q2−1 ≤ 5. In [18] it was shown that 4 k(q) for any odd prime number and in q−1 ≤ (q−1)2 −1 [14] it is proved that k(q) q−2 .

6. Since the automorphism ϕy induced by q- element y is acting on normal subgroup P is regular, sois acting regulary on the set P −{e}, hence qb divides pc − 1. Thus qb ≤ pc − 1.Ifqb = pc − 1 or 2qb = pc − 1 , then from [5] P must be abelian.

7. In [9],[10] it is shown that in case if P is n ot abelian, then pc ≥ 27.

Lemma 2.6 For any prime numbers p and q we have: pq−1 ≡ 1(modq) , i.e. q − 1=min{s ∈ N : ps ≡ 1(modq)} [17]. The nilpotency class of ﬁtting subgroups 701

Theorem 2.7 Let G be a ﬁnite group with basis property which is not a p- group, then the nilpotency class of the Fitting subgroup Fit(G) has no upper bound. Moreover there exists a ﬁnite group with basis property for each prime number q>2, such that: c(Fit(G)) = q − 1, = q. Proof. Let p = q be prime numbers consider the polynomial:

f(x)=xq−1 + xq−2 + ... + x +1∈ GF (p)[x], over the ﬁeld GF (p). By [11] The polynomial f(x) is irreducible over GF (p). We will construct the group satisfying basis property which has the nilpotency class q − 1.Let P = UT(q,GF(pq−1)), i.e., P is the upper triangular group We know from [15] that P is nilpotent group with nilpotency class q − 1 . From [15]. P is Sylow subgroup of the general linear group GF (q,GF(pq−1)), so that P is a p- group.By lemma 2.6 we have q|pq−1 − 1 , hence the multi- plicative group of GF (pq−1) contains a subgroup of order q .Let

q−1 x S = {d1,d2, ..., dq):di ∈ GF (p ) ,i=1, ..., q}

We consider the diagonal matrrix group

H = diag(d1,d2, ..., dq) so |H| = q.LetG =. Then from [5] G =P and the action of H on P gives rise to a regular automorphism with kernel P and complement H . Let us denote this automorphism by

−m m ϕhm : A → h Ah ,A∈ P

We will show that G is satisﬁes basis property. Let hm ∈ H , where m = 1, ..., q − 1 and I = A ∈ P .Put

K = and g(x)=xq−1 + xq−2 + ... + x +1∈ Z[x], Consider the image f(x) of g(x) under the homomorphism

λ : Z[x] → Zp[x],

λ(g(x)) = f(x)=xq−1 + xq−2 + ... + x +1∈ GF (p)[x], 702 M. Alkadhi, A. Al Khalaf and M. Quick f(x) is irreducible polynomial over GF (p) and it divides xq − 1 but does not divide x − 1. By [11] the automorphism ϕhm is regular then we have: q−1 q−2 ϕhm (A)ϕhm (A)ϕhm (A)A = I ∈ φ(K), where φ(K) is the Frattini subgroup of K.LetghmA(x) is the polynomial of operator ϕhm , where A = I, remains the same on all quotient Frattini subgroups, thus: ghmA(x)=g(x), meaning that the representation is isotopic, i.e. the group G satisﬁes the basis property. Corollary 2.8 Let G be a ﬁnite group with basis property and let q be a prime divisor of the order of the complement Frobenius subgroup. If kˆ(q) an upper bound of the nilpotency class of Fit(G),then kˆ(q) ≥ q − 1.

Example 2.9 Let G = A4, then: 1. G satisﬁes the basis property. 2. cF it(G)) = 2.

Let G = A4, then: 2 1. We know that |A4| =12=2 · 3. It is a semidirect product of the 2- group K by the 3-group , where =< (1 2 3) > and K =< (1 2)(3 4), (1 3)(2 4) > it is a 2-group and K A4 We can consider that K is a vector space with dimension 2 over the ﬁeld GF (2). Since K A4, then y induced on K a linear operator:

ϕy : K → K; −1 ϕy : α → y αy, 3 α ∈ K such that ϕy = idK and ϕy = α; α ∈ K \{(i)} We have the equation: 3 2 (ϕy − idK)(α)=(ϕ − idK)(ϕ + ϕ + idK)(α)=0. α ∈ K\{()} Because the operator (ϕ−idK) is regular, then (ϕ−idK) =0 and (ϕ2 + ϕ + idK)(α)=0, and the irreducible polynomial Z2 + Z +1 over the ﬁeld GF (2) is minimal polynomial of the operator ϕ for all nonzero vectors of K.Ifα ∈ K and u ∈ A4 \ K, then the automorphism −1 ϕu : α → u αu is regular and induces an irreducible hence isotopic representation, then conditions in the theorem2.3 for basis property are satisﬁed, hence A4 is group with basis property. 2. To show cF it(G)) = 2, just consider the normal series:

< (i) > K A4, . The nilpotency class of ﬁtting subgroups 703

References

[1] Al Khalaf A. Finite group with basis property, Dok. Acad. Nauk BSSR, n. 11, 1989 (Russian).

[2] Al Khalaf A. Frobenius group with basis property, J. of Al-Baath Un., Syr., v. 13, 1994 (Arabic).

[3] Al Khalaf A. Finite izoordinal group, J. of Al-Baath Un, Syr., v. 20, 1998 (Arabic).

[4] Higman G. Finite groups in which every element has prime power order, J. Lond. Math. Soc., v. 32, 1957,p. 335-342.

[5] Huppert G.Endliche Gruppen I ,Berlin, 1967. 793 p.

[6] Hughes I. Group with ﬁxed-point-free Automorphism, Math. Repts. Acad. Sci Can. v. 7, 1995, p.22-61.

[7] Jones P. A basis theorem for free inverse Semigroup, J. Algebra, v. 49, 1977. p.172-190.

[8] Jones P. Basis properties for inverse Semigroups, J. Algebra, v. 50, 1978. p.135-152.

[9] Kreknin V., Kostrikin A. I. On Lie algebras with regular automorphism, Dokl. Acad. Sci. USSR, 1963 , T. 149, No 2, p. 249- 251 (Russian).

[10] Kreknin V. On solvable Lie algebras with regular automorphism with ﬁnite order, Dokl. Acad. Sci. USSR, 1963 , T. 150, No 3, p. 467- 469 (Russian).

[11] Lidl R. Pilz G.,Applied abstract algebra New York 1984. 543p.

[12] Neumann B.H. Groups with automorphism that leave only the neutral element ﬁxed, Archive, der Math.1956.B.7. s. 1-6.

[13] Suprinenko D.A.Matrix groups,Izd. Nauka,Moscow, 1972 (Russian).

[14] Starostin O.S.On Frobenius groups,Ukr. J., 1971 T. 23, No 5 p. 629 - 639 (Russian).

[15] Steptchuk O.Nonabelian groups with regular automorphism,Tez. Doklad. XIV Un. Alg. Conf., 1981 (Russian).

[16] Suzuki M. Finite groups with nilpotent centralizers, Trans. Amer. Soc., 1961, v. 99, p. 425-470. 704 M. Alkadhi, A. Al Khalaf and M. Quick

[17] Stein W. Elementary number theory a computational approach, Springer- Verlag, 2007.

[18] Thompson J. Finite groups with ﬁxed-point-free Automorphism of prime order,Proc. Math. Acad. Sci. USA, 1959, v. 45, p. 578-581.

Received: February, 2012