Complements to abelian normal subgroup are automorphic

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Statement

Suppose G is a group, N is an Abelian normal subgroup, and H, K are permutable complements to N in G. Then, there is an automorphism \sigma of G such that:

  • The restriction of \sigma to N is the identity map on N.
  • \sigma induces an isomorphism from H to K.

In particular, H and K are automorphic subgroups.

(Note: There may exist automorphic subgroups to H that are not permutable complements to N).

Related facts

Breakdown when the normality constraint is removed or shifted

Other forms of breakdown