Let
PSL
(
n
,
q
)
{\operatorname {PSL}}(n,q)
denote the projective special linear group of degree
n
n
over
GF
(
q
)
{\text {GF}}(q)
, the field with
q
q
elements. The following theorem is proved. Theorem. Let
G
G
be a simple group of order
2
a
3
b
5
c
7
d
p
,
a
>
0
,
p
{2^a}{3^b}{5^c}{7^d}p,a > 0,p
an odd prime. If the index of a Sylow
p
p
-subgroup of
G
G
in its normalizer is two, then
G
G
is isomorphic to one of the groups,
PSL
(
2
,
5
)
,
PSL
(
2
,
7
)
,
PSL
(
2
,
9
)
,
PSL
(
(
2
,
8
)
,
PSL
(
2
,
16
)
,
PSL
(
2
,
25
)
,
PSL
(
2
,
27
)
,
PSL
(
2
,
81
)
{\operatorname {PSL}}(2,5),{\operatorname {PSL}}(2,7),{\operatorname {PSL}}(2,9),{\operatorname {PSL}}((2,8),{\operatorname {PSL}}(2,16),{\operatorname {PSL}}(2,25),{\operatorname {PSL}}(2,27),{\operatorname {PSL}}(2,81)
, and
PSL
(
3
,
4
)
{\operatorname {PSL}}(3,4)
.