The center and the commutator subgroup of Q is the subgroup } . } } The factor group Q / { e, } is isomorphic to the Klein four-group V . The inner automorphism group of Q is isomorphic to Q modulo its center, and is therefore also isomorphic to the Klein four-group.
22.
In their algebraic formulation of quantum mechanics the equation of motion takes on the same form as in the Heisenberg picture, except that the " bra " and " ket " in the bra ket notation each stand for an element of the algebra and that the Heisenberg time evolution is an inner automorphism in the algebra.
23.
Equivalently, a group is quasisimple if it is equal to its commutator subgroup and its inner automorphism group Inn ( " G " ) ( its quotient by its center ) is simple ( and it follows Inn ( " G " ) must be non-abelian simple, as inner automorphism groups are never non-trivial cyclic ).
24.
Equivalently, a group is quasisimple if it is equal to its commutator subgroup and its inner automorphism group Inn ( " G " ) ( its quotient by its center ) is simple ( and it follows Inn ( " G " ) must be non-abelian simple, as inner automorphism groups are never non-trivial cyclic ).
25.
In the first case, the transformation is an inner automorphism, which is a way of expressing the enfolding and unfolding movement in terms of " potentialities " of the process; in the second case it is an outer automorphism, or transformation to a new Hilbert space, which is a way of expressing an " actual change ".
26.
All the corresponding homogeneous spaces are symmetric, since the subalgebra is the fixed point algebra of an inner automorphism of period 2, apart from G 2 / A 2, F 4 / A 2 �A 2, E 6 / A 2 �A 2 �A 2, E 7 / A 2 �A 5 and all the E 8 spaces other than E 8 / D 8 and E 8 / E 7 �A 1.
