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Automorphism of a subnormal series

Definition

Consider a subnormal series of the form:

1 = K_0 \triangleleft K_1 \triangleleft \ldots \triangleleft K_n = G

an automorphism of the subnormal series is an automorphism \sigma of G such that \sigma(K_i) = K_i for every i.

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.