Retract not implies every permutable complement is normal

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Statement

Suppose G is a group and H is a retract of G. In other words, H has a normal complement in G: there exists a normal subgroup N of G such that N \cap H is trivial and NH = G.

There may exist non-normal permutable complements to H in G. In other words, there may exist a non-normal subgroup K of G such that HK = G and H \cap K is trivial.

Related facts

Proof

Example

Further information: symmetric group:S4

Let G be the symmetric group on the set \{ 1,2,3,4 \}, H be the subgroup comprising permutations on \{ 1,2,3 \}. Then:

  • H has a normal complement, namely the subgroup N = \{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}.
  • H has a permutable complement that is not normal, namely the subgroup K = \{ (), (1,2,3,4), (1,3)(2,4), (1,4,3,2)\}.