Stability automorphism of subnormal series: Difference between revisions

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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 $σ$ of a group $G$ is termed a stability automorphism with respect to the subnormal series:

${e} = H_{0} ◃ H_{1} ◃ \dots ◃ H_{n} = G$

if $σ (H_{i} x) = H_{i} x$ for any $x \in H_{i + 1}$ , or equivalently, $σ$ acts as identity on $H_{i + 1} / H_{i}$ .

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

@@ Line 6: / Line 6: @@
 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===