Final steps are done in proving that the groups
SL
(
n
,
Z
)
\operatorname {SL}(n,\mathbb {Z})
,
GL
(
n
,
Z
)
\operatorname {GL}(n,\mathbb {Z})
and
PGL
(
n
,
Z
)
\operatorname {PGL}(n,\mathbb {Z})
are
(
2
,
3
)
(2,3)
-generated if and only if
n
≥
5
n\ge 5
, and
PSL
(
n
,
Z
)
\operatorname {PSL}(n,\mathbb {Z})
is
(
2
,
3
)
(2,3)
-generated if and only if
n
=
2
n=2
or
n
≥
5
n\ge 5
. In particular, the results cover the remaining cases of
n
=
8
n=8
, …,
12
12
, and
14
14
.