The following
2
2
-generator
6
6
-relator presentation is obtained for the
3
3
-dimensional special linear group
SL
(
3
,
Z
k
)
\operatorname {SL}(3,{\mathbb {Z}_k})
for each odd integer
k
>
1
k > 1
:
\[
SL
(
3
,
Z
k
)
=
⟨
x
,
y
|
x
3
=
y
3
=
(
x
y
)
6
=
(
x
−
1
y
x
−
1
y
−
1
x
y
)
2
=
(
x
y
−
1
x
y
x
y
−
1
x
−
1
y
−
1
)
k
=
(
(
x
y
−
1
x
y
x
y
−
1
x
−
1
y
−
1
)
(
k
−
1
)
/
2
x
y
)
4
=
1
⟩
.
\operatorname {SL}(3,{\mathbb {Z}_k}) = \langle x,y|{x^3} = {y^3} = {(xy)^6} = {({x^{ - 1}}y{x^{ - 1}}{y^{ - 1}}xy)^2} = {(x{y^{ - 1}}xyx{y^{ - 1}}{x^{ - 1}}{y^{ - 1}})^k} = {({(x{y^{ - 1}}xyx{y^{ - 1}}{x^{ - 1}}{y^{ - 1}})^{(k - 1)/2}}xy)^4} = 1\rangle .
\]
Alternative presentations for these groups and other groups associated with them are also given.