On automorphisms of extremal even unimodular lattices of dimension 48

The automorphism groups of the three known extremal even unimodular lattices of dimension 48 and the one of dimension 72 are computed using the classification of finite simple groups. Restrictions on the possible automorphisms of 48-dimensional extremal lattices are obtained. We classify all extremal lattices of dimension 48 having an automorphism of order $m$ with $\phi (m)>24$. In particular the lattice $P_{48n}$ is the unique extremal 48-dimensional lattice that arises as an ideal lattice over a cyclotomic number field.