Certain classes of group presentations

Abstract

In Chapter two we look at the class F(n) = <R, S | Rⁿ = Sⁿ = (Rᵃ¹Sᵇ¹)ˣ¹(Rᶜ¹Sᵈ¹)ʸ¹(Rᵃ²Sᵇ²)ˣ² (Rᶜ²Sᵈ²)ʸ² …(RᵃᵐSᵇᵐ)ˣᵐ (RᶜᵐSᵈᵐ)ʸᵐ = 1 >. For some values of n, a[sub]i , b[sub]i, d[sub]i, x[sub]i, y[sub]i we give results on these groups where we have been able to determine their order, either finite or infinite. In the last section in Chapter two we study two classes of groups generated by A and B and subject to the following relations:

Relations for class 1:

A⁴ = 1, B⁴ = 1, (B(AB)²)⁴ = 1, (B(BA)⁶)⁴ = 1, (B(BA)¹⁴)⁴ = 1, …, B(BA)⁽²⁽ⁿ⁻¹⁾ᐟ²-2)⁴ = 1 A⁻¹B⁻¹)²⁽ⁿ⁻³⁾ᐟ²B(BA)⁽²⁽ⁿ⁻¹⁾ᐟ²-2)B(BA)⁽²⁽ⁿ⁻³⁾ᐟ²B⁻¹(A⁻¹B⁻¹)²⁽ⁿ⁻¹⁾ᐟ²-2) B⁻¹ A⁻¹B⁻¹)²⁽ⁿ⁺¹⁾ᐟ²-3) A(BA)⁽²⁽ⁿ⁻¹⁾ᐟ²-1)B⁻¹= 1 (BA)²⁽ⁿ⁻¹⁾ᐟ² B⁻¹(A⁻¹B⁻¹)²⁽ⁿ⁻¹⁾ᐟ²-2) B⁻¹(A⁻¹B⁻¹)²⁽ⁿ⁺¹⁾ᐟ²-3) A² =1

Relations for class 2:

A⁴ = 1, B⁴ = 1, (B(AB)²)⁴ = 1, (B(BA)⁶)⁴ = 1, (B(BA)¹⁴)⁴ = 1, …, B(BA)⁽²⁽ⁿᐟ²⁻²⁾)⁴ = 1 , B⁻¹(BA)² ⁽ⁿ⁻²⁾ᐟ²B(BA) ⁽²ⁿᐟ²⁻²⁾ B(A⁻¹B⁻¹)²⁽ⁿ⁻²⁾ᐟ²-1) = 1, (BA) ⁽²ⁿᐟ²+2⁽ⁿ⁻²⁾ᐟ²+2)B(BA) ⁽²ⁿᐟ²-2)B(A⁻¹B⁻¹)²⁽ⁿ⁻²⁾ᐟ²-1)A² =1.

The groups in the first class turn out to be the cyclic group of order 2 and the groups in the second class turn out to be metabelian groups of order 4. (2ⁿᐟ²-1)² . Moreover the derived group of the groups in the second class is the direct product of two copies of a cyclic group of order (2ⁿᐟ²-1)². In Chapter three we study the groups with a presentation of the form:

<A,B|A⁴ = 1, Bⁿ = 1, AⁱBʲAᵏBᵗ =1

and determine all possibilities with conditions: j+t = 0 and i,k ∊ { + 1, 2 }.

Also in the second section of Chapter three we study the groups with a presentation of the form:

<A,B | A⁴ = 1, Bⁿ =1, AⁱBʲAᵏBᵗA ᵐBᵖ =1> and determine some of the possibilities with conditions: j = l,t = l,p = -2 and i,k,m ∊ ℤ. In Chapter four we give new efficient presentations for the groups PSL(2,p), where p is an odd prime, p ∊ { 5,7,11,13,17,19,23,29,31,37, 41,43,53,59,79,83,89,109,139,229 }. We give permutation generators for these groups which satisfy our efficient presentation. Also we give new efficient presentations for PSL(2,p), where p is a prime power and p ∊ { 9,25,27,49,169}. Also in Chapter four, permutation generators are given for these groups which satisfy our presentations. In Chapter five we give new efficient presentations for the groups SL(2,p), where p is an odd prime and p ∊ { 5,7,11,13,17,19,23,29,31,41, 43,53,79,89,109,139,229 }. Also we give new efficient presentations for the groups SL(2,p), where p is an prime power and p ∊ { 8,16,25,27,49,169 }. In Chapter six we study the class of groups with the presentation

<a,b |aᵖ =1, bᵐ⁺ᵖa⁻ᵐbᵐa⁻ᵐ =1, (ab)² = 1>

,p an odd number and m ∊ ℤ. For some values of p and m these groups have connections with the groups PSL(2,p). In Chapter 7 we attempt to show the efficiency of PSL(2, ℤ[sub]n ) x PSL(2, ℤ[sub]m). For some values of n and m we give efficient presentation for these groups. In the same chapter we also attempt to show the efficiency of PSL(2, ℤ [sub]p) x PSL(2,32). For some values of p we give an efficient presentation for these groups. In the last section of the thesis we give efficient presentations for the following direct products

(i) PSL(2,5) X PSL(2,3²) (ii) PSL(2,7) X PSL(2,3²) (iii) PSL(2,5) X PSL(2,3³)

Also in the last section of the thesis the structure of a perfect group of order 161280 is investigated.