C. M. Campbell

ON PRESENTATIONS OF PSL(2,p n )

We give presentations for the groups PSL (2, p n ), p prime, which show that the deficiency of these groups is bounded below. In particular, for p = 2 where SL (2, 2 n ) = PSL (2, 2 n ), we show that these groups have deficiency greater than or equal to – 2. We give deficiency – 1 presentations for direct products of SL (2, 2 n ) for coprime n i . Certain new efficient presentations are given for certain cases of the groups considered.

FINITE GROUPS OF DEFICIENCY ZERO INVOLVING THE LUCAS NUMBERS

In this paper, we investigate a class of 2-generator 2-relator groups G ( n ) related to the Fibonacci groups F(2, n ), each of the groups in this new class also being defined by a single parameter n , though here n can take negative, as well as positive, values. If n is odd, we show that G ( n ) is a finite soluble group of derived length 2 (if n is coprime to 3) or 3 (otherwise), and order |2 n ( n + 2) g n f ( n , 3) |, where f n is the Fibonacci number defined by f 0 =0, f 1 =1, f n +2= f n + f n+1 and g n is the Lucas number defined by g 0 = 2, g 1 = 1, g n +2 = g n + g n +1 for n ≧0. On the other hand, if n is even then, with three exceptions, namely the cases n = 2,4 or –4, G(n) is infinite; the groups G (2), G (4) and G (–4) have orders 16, 240 and 80 respectively.

REMARKS ON A CLASS OF 2-GENERATOR GROUPS OF DEFICIENCY ZERO

Let G be a finitely presented group. A finite presentatio 0*n of G is said tohave deficiency — m n if it defines G with m generators and n relations. Thedeficiency of G is the maximum of the deficiencies of all the finite presentations &of G. If G is finite the deficiency o Gf is less than or equal to zero. The only finitetwo generator groups of deficiency zero that are known are certain metacyclicgroups given by Wamsley (1970), a class of nilpotent groups given by Macdonaldin (1962) and a class of groups given by Wamsley (1972).In this paper we consider a class of two generator groups of deficiency zero.Define the group G(m, n), where m, n are non-zero integers, byG(m,n) = |[a

Finite one-relator products of two cyclic groups with the relator of arbitrary length

In this paper we consider the groups G = G(α, n) defined by the presentations . We derive a formula for [ G ′: ″ ] and determine the order of G whenever n ≦ 7. We show that G is a finite soluble group if n is odd, but that G can be infinite when n is even, n ≧ 8. We also show that G (6, 10) is a finite insoluble group involving PSU (3, 4), and that the group H with presentation is a finite group of deficiency zero of order at least 114,967,210,176,000.

One-Relator Products of Cyclic Groups and Fibonacci-Like Sequences

In this paper, we mention some properties of certain generalized Fibonacci sequences we have looked at while investigating one-relator products of cyclic groups. The particular groups we have investigated are those defined by presentations of the form rnrn

On the efficiency of some direct powers of groups

rnrn, where R(a, b) is a word of the form ab j (1)ab j (2)…ab j (r) with r ≥ 2 and 0<j(i)<n for each i. Such a group is called a one-relator product of the cyclic groups C2 and Cn of orders 2 and n respectively, in that it is formed from the free product of C2 and Cn by imposing the single extra relation R(a, b) = 1. We denote this group by G(n; j(1), j(2),…, j(r)).

Group presentations where the relators are proper powers

1. Introduction. A finite group G is efficient if it has a presentation on n generators and n +m relations, where m is the minimal number of generators of the Schur multiplier M(G) of G. The deficiency of a presentation of G is r-n, where r is the number of relations and n the number of generators. The deficiency of G, def G, is the minimum deficiency over all finite presentations of G. Thus a group is efficient if def G = m. Both the problem of efficiency and the converse problem of inefficiency have received considerable attention recently; see for example [1], [3], [14] and [15]. In particular, in response to a question of Wiegold [19] concerning the efficiency of direct powers of groups, several papers have appeared; see [4], [5], [6], [12] and [13]. An efficient presentation for PSL(2,5) 2 was given by Kenne in [12] and for PSL(2,5) 3 by Campbell, Robertson and Williams in [5]. In [6], direct squares of the projective special linear group PSL(2,p) are shown to be efficient for all primes p. In addition to considering direct powers, the efficiency of direct products of some simple groups has been investigated; see [11], [13], where efficient presentations are given for PSL(2,5) X A6 and PSL(2,5) x A7. In this paper we give a new method for obtaining efficient presentations of direct products. This method allows us to give a simpler proof to that given in [6] that PSL(2,p)XPSL(2,p) is efficient. The power of this new method is illustrated, in this case, by leading naturally to an efficient presentation for SL(2,p) x PSL(2,p). The main result of this paper uses this method to obtain an efficient presentation for PSL(2,p) x PSL(2,p) x PSL(2,p), where p is a prime. This leaves us a long way short of answering Wiegolds question in [19] which, in essence, asks whether PSL(2,p) n is efficient for all n. However, it is worth noting that the approach given here gives an efficient presentation for PSL(2,p) 3 on six generators that looks more promising than earlier methods which relied heavily on obtaining two-generator presentations. We give two further applications of the method, showing that A6 X PSL(2,p) and PSL(2,p) x / are efficient for all primes p,p>5, where J = PSL(2, q) x C2 or / = PGL(2, q) depending on q. Finally we give an efficient presentation for PSL(2,5) For any group G, we let G denote the commutator subgroup of G and let Z{G) denote the centre of G. The group H is a central extension of G if there is a subgroup Z == Z(H) with H/Z ^ G. H is irreducible if there is no L < H with H = ZL. In the case that Z<// , H is called a stem extension of G and a stem extension with Z = Af(G) is called a covering group of G. For perfect groups G, a covering group is uniquely determined. Moreover, if G is perfect and H is a central extension of G, then H is a perfect irreducible central extension of G with HI{Z f~l H) = G; that is, H is a stem extension of G. For further information on stem extensions see [17]. The Schur multiplier of the direct product G x H may be calculated by the Schur-KUnneth formula M(GxH) = M(G)xM(H)x(G®H); see [18]. The Schur