An inner automorphism [theta] is one such that there exists u [member of] U(M) for which [theta](x) = uxu* for all x [member of] M.
We have just shown that all inner automorphisms of [??] are liftable to inner automorphisms of M.
[??]--the inner automorphism of G, generated by an element g [member of] G;
Then the factor-group G/Z(G) is non-cyclic, i.e., [absolute value of G/Z(G)] [greater than or equal to] 4 and G has at least 4 inner automorphisms. Therefore, [absolute value of Aut(G)] [greater than or equal to] 4.
Consider now the inner automorphism of R induced by the invertible matrix P = I + [e.sub.rj], for r [not equal to] i, j: [phi](x) = [P.sup.-1]xP.
On the other hand, if [chi] is the inner automorphisms induced by the invertible matrix Q = I + [e.sub.ri], as above [chi](a)x[chi](q) + [chi](c)x[chi](b) = 0, for all x [member of] R.
This structure is not just a G-action on A, as one might naively expect, but a weak G-action, which is an action by Hopf algebra automorphisms [[phi].sub.g] : A [right arrow] A such that [[phi].sub.g] [omicron] [[phi].sub.h] equals [[phi].sub.gh] only up to an inner automorphism
An element c generates an inner automorphism
[??] for which [[??].sup.4] = 1:
A special case of this result was already obtained in , without any reference to orthogonality and for automorphisms [[phi].sub.h]'s which are coupled with the [e.sub.h]'s in the following sense: [[omega].sub.h] is the inner automorphism
associated to [e.sub.h], with the additional assumption that [e.sub.0] = 1.
Lau and Paterson  gave a necessary condition on a locally compact group G to have an inner invariant mean m such that <m, [1.sub.V]> = 0 for some compact neighborhood V of G invariant under the inner automorphisms
. Let A be a Borel subset of G.