1.15 If m s 35(mod99) and n = 2(mod9) then

G i5“ < a,b I a9(ab)"“^ = 1 , ~ a(^m-2n+38) > g SL(2,109) and this is an efficient presentation for Gjg.

Modifications which have to be made in Theorem 5.1.1 are as follows: 1. If m s 35(mod99) and n = 2(mod9) then G1 5 = G1 5 ,

2. From relation 1 it can be easily seen that dP commutes with b . 3. If m 5 35(mod99) and n s 2(mod9) then a^GZ(G^g )flG ^5.

4. Hi5 = < a9 >.

5. G j5/<a^> s PSL(2,109), by Theorem 4.1.18 .

Theorem 5.1.16 If m s 30(modll5) and n = 2(mod5) then

G j6 ~ < a,b I a^(ab)“^ = 1 , b‘”'*‘^^a'“”b"^a”" = aGm-2n+34) > g SL(2,139) and this is an efficient presentation for G^^.

Modifications which have to be made in Theorem 5.1.1 are as follows: 1. If m s 30(modl 15) and n s 2(mod5) then Gig = G ig .

2. From relation 1 it can be easily seen that a^ commutes with b . 3. If m s 30(m odll5) and n s 2(mod5) then a^eZ(Gj^ )flG jg.

4. H i g= <a 5 > .

5. G i6/<a^> s PSL(2,139). by Theorem 4.1.19 .

Theorem 5.1.17 If m s 52(mod95) and n = 7(mod 19) then

Gjy= < a,b I a^^(ab)-^ = 1, b"^+^a-"b"^a-» = a(^7m-2n+42) > ^ SL(2,229) and this is an efficient presentation for Gjy.

Modifications which have to be made in Theorem 5.1.1 are as follows: 1. If m s 52(mod95) and n s 7(mod 19) then G1 7 - G1 7.

2. From relation 1 it can be easily seen that a^9 commutes with b . 3. If m s 52(mod95) and n s 7(modl9) then a^^eZ(Gjy )flG . 4. Hiy = < al9 >

5.2 DIFFERENT EFFICIENT PRESENTATIONS FOR THE GROUPS SL(2,p«) where p"e { 8,16,25,27,49,169 }:

In general the efficiency of SL(2,p") hasn't been solved yet. In the literature we can find nothing, except for a few cases of particular values of p and n. In [6] C.M.Campbell and E.F.Robertson have given efficient presentation for SL(2,8) and in [8] by the same authors the efficiency of SL(2,16) is shown. The efficiency of SL(2,25), SL(2,27), SL(2,32), SL(2,49) and SL(2,64) is also shown in [9]. In 1988 in [11] the efficiency of SL(2,169) was given by C.M.Campbell and E.F.Robertson.

In this section we study groups of the following types

(i) < a,b I a”^(ab)^ = 1 , ybSm+l > r dependent on m and n (ii) < a,b I a""^(ab)^ = 1 , b‘^‘^^a"“”b”^a‘"'^ = a^ > r is dependent on m and n.

In this section we are going to show that for certain values of k,t,ra,n and r these groups are efficient and isomorphic to the groups SL(2,p").

Theorem 5.2.1 If m e Z and n s 2(mod7) then

< a,b i a^(ab)-^ = 1 , b(63m+ll)a‘“nb(63m+2)a~n = a(315m-2n+32) > g SL(2,8) and this is an efficient presentation for Gj.

Proof: If m e Z and n s 2(mod7) then Gi = G i and consequently a^,

b9 e Gi = G1 . From relation 1 it can be seen that dP commutes with b. Using this

information with relation 2 it can be seen that b9 commutes with a^. Again using relation 1

2p - (ab) 2 a6 as bab

a V = babb^ (**).

From (*) and (**) it can be seen that b^ commutes with a. Since a^, b9 € Gi == G 1 then a^ , b^ e Z(Gi )CiG\ .

Now consider the factor group

Gi/<a^,b9 > s < a,b 1 a^= (ab)^ = (b^a“^)^ = b9= 1 >

Using the CAYLEY program 1 it can be seen that G^/<a7,b9 > is isomorphic to SL(2,8). In the CAYLEY program 1 the presentation for SL(2,8) has been taken from C.M.Campbell and E.F.Robertson [6]. Since Gj/<a^,b9 > = SL(2,8) therefore it can be deduced that <a7,b9 > < M(SL(2,8)). But the Schur multiplier of SL(2,8) is

trivial. This means I <a^,b9 > I = 1. So a^, b^ must be equal to identity element of Gj and consequently Gj = SL(2,8). Since the Schur multiplier of G^ is trivial this yields the claimed result.

Theorem 5.2.2 If m s 4(modl5) and n s 2(mod5) then

G2 = < a,b 1 a^(ab)-^ = 1 , bm+lSa-nym^-n = a(3m-2n+22) > = SL(2,16) and this is an efficient presentation for G2 .

Proof: If m s 4(modl5) and n s 2(mod5) then G2 = G 2 and consequently a^, b^^ e G2 = G 2 . From relation 1 it can be seen that a^ commutes with b . Using this

information with relation 2 it can be seen that b^^ commutes with a?. Again using relation 1

a^ = (ab)^

sâ = bab

b^^a^ = b^^bab... (*) a^b^^ sa babb^^ (**)

From (*) and (**) it can be seen that eventually b^^ commutes with a. Since a^, b^^e G2 = G 2 therefore a^, b^^ e Z(G2 )riG 2 •

Now consider the factor group

G2/<a^,b^^ > s < a,b I a^= (ab)^ = (b'^a"“^)^ = b^^= 1 > Using the CAYLEY program 1 it can be seen that G^/<a^,b^^ > is isomorphic to SL(2,16). In the CAYLEY program 1 the presentation for SL(2,16) has been taken from C.M.Campbell and E.F.Robertson [6]. Since G2/<a^,b^^ > = SL(2,16) therefore it can be deduced that <a^,b^^ > < M(SL(2,16)). But the Schur multiplier of SL(2,16) is trivial. This means I <a^,b^^ > 1 = 1 . So a^, b^^ must be equal to the identity element of G2 and consequently G2 = SL(2,16). Since the Schur multiplier

of G2 is trivial the result follows as claimed.

In the following four theorems full proofs are essentially the same as the proof of Theorem 5.1.1 with slight modifications. In every case instead of full proofs only the modifications will be given which have to be made in the proof of Theorem 5.1.1 in order to obtain the full proof. These modifications are

1. The conditions which makes Gi perfect. 2. The proof that a^ commutes with b. 3. The element a*’ which is in Z(Gj )riG ^.

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

5. The related Theorem 4.2.i which is going to be used in showing, that G j/<aO s PSL(2,p).