  Two mappings in a finite field, the Frobenius mapping and the cyclic shift
mapping, are applied on lines in PG($n,p$) or codes in the Grassmannian, to
form automorphisms groups in the Grassmanian and in its codes. These
automorphisms are examined on two classical coding problems in the
Grassmannian. The first is the existence of a parallelism with lines in the
related projective geometry and the second is the existence of a Steiner
structure. A computer search was applied to find parallelisms and codes. A new
parallelism of lines in PG(5,3) was formed. A parallelism with these parameters
was not known before. A large code which is only slightly short of a Steiner
structure was formed.
