Complemented characteristic not implies left-transitively complemented normal
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 characteristic subgroup) need not satisfy the second subgroup property (i.e., left-transitively complemented normal subgroup)
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Statement with symbols
Example of the dihedral group of order sixteen
Further information: Dihedral group:D16
Let be a dihedral group of order . In other words, is given by the presentation:
Consider the subgroups:
- is complemented characteristic in : is a dihedral group of order eight, with as the unique cyclic normal subgroup of order four, hence is characteristic in . Further, is a complement to in .
- is complemented normal in : Since has index two, it is normal in (subgroup of index two is normal, or alternatively, nilpotent implies every maximal subgroup is normal). Further, the two-element subgroup is a permutable complement to in .
- is not complemented normal in : If were complemented normal in , it would also (by fact (1)) be complemented normal in the intermediate subgroup , which is a cyclic group of order eight. But we know that is not complemented in , since any element of not in generates .