# Stability automorphism of subnormal series

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## Definition

### Symbol-free definition

An automorphism of a group is said to be a **stability automorphism** with respect to a subnormal series if it induces the identity map on each successive quotient for the subnormal series.

The stability automorphisms of any fixed subnormal series form a group, called the **stability group** of that subnormal series. This group lives as a subgroup of the automorphism group.

### Definition with symbols

An automorphism of a group is termed a **stability automorphism** with respect to the subnormal series:

if for any , or equivalently, acts as identity on .

(An analogous definition can be given for subnormal series indexed by infinite sets).