The
mod
p
\bmod \;p
v
1
{v_1}
-periodic homotopy groups of a space
X
X
are defined by considering the homotopy classes of maps of a Moore space into
X
X
and then inverting the Adams self map. In this paper we compute the
p
p
v
1
{v_1}
-periodic homotopy groups of an odd dimensional sphere, localized at an odd prime. This is done by showing that these groups are isomorphic to the stable
mod
p
\bmod \;p
v
1
{v_1}
-periodic homotopy groups of
B
Σ
p
2
(
p
−
1
)
n
B\Sigma _p^{2(p - 1)n}
, the
2
(
p
−
1
)
n
2(p - 1)n
skeleton of the classifying space for the symmetric group
Σ
p
{\Sigma _p}
. There is a map
Ω
2
n
+
1
S
2
n
+
1
→
Ω
∞
(
J
∧
B
Σ
p
2
(
p
−
1
)
n
)
{\Omega ^{2n + 1}}{S^{2n + 1}} \to {\Omega ^\infty }(J \wedge B\Sigma _p^{2(p - 1)n})
, where
J
J
is a spectrum constructed from connective
K
K
-theory, and the image in homotopy is studied.