E. F. Robertson

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.

Efficient presentations for certain simple groups

In this paper we give 2-generator 3-relation presentations for the simple groups PSU(3,5) of order 126000 and the Mathieu group M22Zof order 443520. We also give a 2-generator 2-relation presentation for the Janko group J1of order 175560. This proves that these three simple groups are efficient.

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

Certain one-relator products of semigroups

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)).