# Automorphism of a subnormal series

## 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.