Let
F
\mathfrak {F}
be a semisimple Jordan algebra over an algebraically closed field
Φ
\Phi
of characteristic zero. Let
G
G
be the automorphism group of
F
\mathfrak {F}
and
Γ
\Gamma
the structure groups of
F
\mathfrak {F}
. General results on
G
G
and
Γ
\Gamma
are given, the proofs of which do not involve the use of the classification theory of simple Jordan algebras over
Φ
\Phi
. Specifically, the algebraic components of the linear algebraic groups
G
G
and
Γ
\Gamma
are determined, and a formula for the number of components in each case is given. In the course of this investigation, certain Lie algebras and root spaces associated with
F
\mathfrak {F}
are studied. For each component
G
i
{G_i}
of
G
G
, the index of
G
G
is defined to be the minimum dimension of the
1
1
-eigenspace of the automorphisms belonging to
G
i
{G_i}
. It is shown that the index of
G
i
{G_i}
is also the minimum dimension of the fixed-point spaces of automorphisms in
G
i
{G_i}
. An element of
G
G
is called regular if the dimension of its
1
1
-eigenspace is equal to the index of the component to which it belongs. It is proven that an automorphism is regular if and only if its
1
1
-eigenspace is an associative subalgebra of
F
\mathfrak {F}
. A formula for the index of each component
G
i
{G_i}
is given. In the Appendix, a new proof is given of the fact that the set of primitive idempotents of a simple Jordan algebra over
Φ
\Phi
is an irreducible algebraic set.