Difference between revisions of "Automorphism of a subnormal series"
From Groupprops
m (1 revision) 

(No difference)

Latest revision as of 22:51, 7 May 2008
Definition
Consider a subnormal series of the form:
an automorphism of the subnormal series is an automorphism of such that for every .
The automorphism group of a subnormal series is thus a subgroup of the automorphism group of the whole group.
Facts
 Any automorphism of a subnormal series, gives rise naturally to an automorphism of the associated direct sum. This defines a homomorphism from the automorphism group for the subnormal series to the automorphism group of the associated direct sum. The kenrel of the homomorpihsm is precisely the stability group.
 When the subnormal series is a normal series, then any inner automorphism lives inside the automorphism group.
 When the subnormal series is a central series, then any inner automorphism lives inside the stability group.