2.1 If m s l(mod24 ) and n = 1 then

Fj = < a,b I a^ = (ab)^ = b™+^a""(b"’a“'’)‘’ = 1 > s PSL(2,9) and this is an efficient presentation for Fj.

Proof: If m s l(mod24) and n = 1 then Fi = F i and consequently b^eFj ~ F j- From relation 3 and using relation 1 it can be easily seen that b^ commutes with a, so b"^€Z(Fj). Since b'^eF j it can be noticed that b^eZ(Fj )flF j.

Now consider the homomorphic image of F j by Hi = < b*^ > in other words Fj/<b'^ > = < a,b I a^ = (ab)^ = (ba~^)^ - b^ = 1 >.

Now take K i= < a~^ba"*^,b“ ^ab"^a > < Fj/<b'^ >

Using CAYLEY program 1 it can be seen that permutation representation of

F2/<b'^> on the cosets of K| is isomorphic to PSL(2,9) group.

Using TC on L. == < a > < Fj/<b'^ > it can be verified that the order of

Fj/<b'^ > is equal to the order of the group PSL(2,9). Hence F^/<b^ > = PSL(2,9). Since b^eZ(Fj )flF j it can be deduced that <b^> < M(PSL(2,9)) = C g. This means 1 <b^ > I = 1 or 2 or 3 or 6. In any case b^^ = 1 holds in Fj . Adding this

relation into the group and using the fact that m s l(mod24) and b'^eZ(Fi) the following can be deduced

Fj = < a,b I a^ = (ab)^ = b^a~^(ba” ^)^ = b^*^ = 1 >

Again using TC on L. = < a > it can be seen that I Fj I = I Fj/<b^ > 1, therefore FjS PSL(2,9).

Fj has 2 generators, 3 relations and the Schur multiplier of F j is so this is an efficient presentation for F j .

Theorem 4.2.2 If m s 2(mod5), m 4= 65t +12 and n ~ 5 then

< a,b 1 = (ab)2 = b ^ + ^ a -V a ”” = 1 > s PSL(2,25) and this is an efficient presentation for F2 .

Proof: If m s 2(mod5), m 4* 65t + 12 and n = 5 then F2 — F 2 and consequently

b ^ sF2 ~ F 2* From relation 3 and using relation 1 it can be easily seen that b^

commutes with a^ . Using this fact and the fact that 5 is coprime to 13, the order of a, b^ commutes with a therefore b^eZ(F2). Since b^eF 2 it can be noticed that

b5eZ(F2 )nF2.

Now consider the homomorphic image of F2 by Hi = < b^ > in other words F2/<b^ > = < a,b I a^^ = (ab)^ = (b^a“^)^ = b^ = 1 >

Now take K i= < a” ^ba“ ^a“^b“^a~^b’“ V b “^a > < F2/<b^ >

Using the CAYLEY program 1 it can be seen that the permutation representation of F2/<b^> on the cosets of Kj is isomorphic to the group PSL(2,25).

Using TC on Lj = < a > < F2/<b^ > it can be verified that order of F2/<b^> is equal to the order of the group PSL(2,25). Hence F2/<b^ > £ PSL(2,25).

Since b^eZ(F2 )riF2 it can be deduced that <b^> < M(PSL(2,25)) = C2 . This means I <b^ > I = 1 or 2. i.e.

F2 = PSL(2,25) or F2 is isomorphic to its covering group, SL(2,25). In either case b^® = 1 holds in F2 .

Adding this relation to the group and using the fact that

m = 2(mod5), m 4= 65t + 12 and b^eZ(F2), the following can be deduced:

F2 = < a,b I a^^ = (ab)^ = (b^a“ ^)^ = b^^= 1 >

Again using TC on L. = < a > it can be seen that I F2 I = I F2/<b^ > I. Therefore F2 = PSL(2,25).

p2 has 2 generators, 3 relations and the Schur multiplier of ? 2 is C2 so this is

an efficient presentation for F2 .

In the following three theorems full proofs are essentially the same as the

proof of Theorem 4.2.2 with modifications. In every case instead of full proofs only the modifications will be given which have to be made in the proof of Theorem 4.2.2 and in the CAYLEY program 1, in order to obtain the full proof. These modifications

are:

1. The conditions which makes Fi perfect. 2. The proof that b^ commutes with a. 3. The element b^ which is in Z(Fj )flF j .

4. The subgroup Hi which is going to be used to construct the homomorphic image F/<b^ >.

5. The subgroup Ki which is going to be used to construct the permutation representation of Fj/<b^ > on the cosets of K i.

6. The subgroup Li which is going to be used to enumerate Fj/<b^ > and

F;/<bZr >.

7. The conditions for card(s) and card(t) in the CAYLEY program 1.

Additionally it can be pointed out that, actually Ki is a maximal subgroup of Fi

with minimal index. Therefore with the CAYLEY program 2, a permutation representation has been obtained for these Fi groups. In every case we give these permutation generating pairs.

For Theorem 4.2.1 and Theorem 4.2.2 the permutation generating pairs are respectively of degree 6 and 26 as follows :

(3 = (1,3,2,4)(5,6).

« = (1,2,22,21,14,15,16,17,13,18,19,20,3X4,23,26,24,8,9,10,11,12,25,7,6,5). p = (1,9,25,19,23)(2,4,5,10,3)(6,16,26,18,11)(7,8,24,15,17)(12,13,14,22,20).

In the CAYLEY program 2, for both theorems, card(s) and card(t) must be greater than 1.

Theorem 4.2.3 If m s 3(modl3), m 4= 9 It + 29 and n = 3 then

p3 = < a,b

I

The permutation generating pair of degree 28 is: « = (1,2,4,5,7,6,3)(8,21,20,19,14.17,9)(10,11,12,13,18,15,16) (22,26,28,27,25,23,24),

P = (1,15,14,13,10,4,6,5,16,3,2,9,8)(11,12,23,22,20,27,26,25,21,17,18,19,24). In the CAYLEY program card(s) and card(t) must be greater than 1.

Modifications which have to be made in Theorem 4.2.2 are as follows: 1. If m = 3(modl3), m 4= 91t + 29 and n = 3 then F3 = F3.

2. From relation 3 and using relation 1 it can be easily seen that bl3 commutes with a^ . Using relation 1 it can be seen that b^^ commutes with a.

3. If m s 3(modl3), m 4= 91t + 29 and n = 3 then bl3eZ(Fg)riF3.

4. H 3 = < bl 3 > .

5. K3 = < a'”^ba“ ^,a“ ^b“"^aba“ ^b~ ^a >. 6. L3= < b>.