Complemented normal not implies direct factor

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This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., complemented normal subgroup) need not satisfy the second subgroup property (i.e., direct factor)
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Statement

It is possible to have a group G and a complemented normal subgroup H such that H is not a direct factor of G. In other words, H has a permutable complement in G, but no normal complement in G.

Examples

Here are some examples where the ambient group is quite important and easy to understand:

 Group partSubgroup partQuotient part
A3 in S3Symmetric group:S3Cyclic group:Z3Cyclic group:Z2

Here are some examples where the ambient group is somewhat more complicated:

 Group partSubgroup partQuotient part
A4 in S4Symmetric group:S4Alternating group:A4Cyclic group:Z2
Cyclic maximal subgroup of dihedral group:D8Dihedral group:D8Cyclic group:Z4Cyclic group:Z2
Klein four-subgroup of alternating group:A4Alternating group:A4Klein four-groupCyclic group:Z3
Klein four-subgroups of dihedral group:D8Dihedral group:D8Klein four-groupCyclic group:Z2
Normal Klein four-subgroup of symmetric group:S4Symmetric group:S4Klein four-groupSymmetric group:S3

Here are some examples where the ambient group is even more complicated:

 Group partSubgroup partQuotient part
Cyclic maximal subgroup of dihedral group:D16Dihedral group:D16Cyclic group:Z8Cyclic group:Z2
Cyclic maximal subgroup of semidihedral group:SD16Semidihedral group:SD16Cyclic group:Z8Cyclic group:Z2

Proof

Further information: dihedral group:D8, cyclic maximal subgroup of dihedral group:D8

We can take G to be the dihedral group D_8 given as:

\langle a,x \mid a^4 = x^2 = e, xax = a^{-1} \rangle

and H to be its unique cyclic maximal subgroup \langle a \rangle. Then \langle x \rangle is a permutable complement to H in G, but H has no normal complement, so is not a direct factor.