We investigate the extent to which the exchange relation holds in finite groups
G
G
. We define a new equivalence relation
≡
m
\equiv _{\mathrm {m}}
, where two elements are equivalent if each can be substituted for the other in any generating set for
G
G
. We then refine this to a new sequence
≡
m
(
r
)
\equiv _{\mathrm {m}}^{(r)}
of equivalence relations by saying that
x
≡
m
(
r
)
y
x \equiv _{\mathrm {m}}^{(r)}y
if each can be substituted for the other in any
r
r
-element generating set. The relations
≡
m
(
r
)
\equiv _{\mathrm {m}}^{(r)}
become finer as
r
r
increases, and we define a new group invariant
ψ
(
G
)
\psi (G)
to be the value of
r
r
at which they stabilise to
≡
m
\equiv _{\mathrm {m}}
.
Remarkably, we are able to prove that if
G
G
is soluble, then
ψ
(
G
)
∈
{
d
(
G
)
,
\psi (G) \in \{d(G),
d
(
G
)
+
1
}
d(G) +1\}
, where
d
(
G
)
d(G)
is the minimum number of generators of
G
G
, and to classify the finite soluble groups
G
G
for which
ψ
(
G
)
=
d
(
G
)
\psi (G) = d(G)
. For insoluble
G
G
, we show that
d
(
G
)
≤
ψ
(
G
)
≤
d
(
G
)
+
5
d(G) \leq \psi (G) \leq d(G) + 5
. However, we know of no examples of groups
G
G
for which
ψ
(
G
)
>
d
(
G
)
+
1
\psi (G) > d(G) + 1
.
As an application, we look at the generating graph
Γ
(
G
)
\Gamma (G)
of
G
G
, whose vertices are the elements of
G
G
, the edges being the
2
2
-element generating sets. Our relation
≡
m
(
2
)
\equiv _{\mathrm {m}}^{(2)}
enables us to calculate
A
u
t
(
Γ
(
G
)
)
\mathrm {Aut}(\Gamma (G))
for all soluble groups
G
G
of nonzero spread and to give detailed structural information about
A
u
t
(
Γ
(
G
)
)
\mathrm {Aut}(\Gamma (G))
in the insoluble case.