example of non-permutable subgroup
As the group is nilpotent we know every subgroup is subnormal; however, not every subgroup is permutable. In particular, observe that for two general subgroups and of , it may be possible that is not a subgroup. In this situation we find our counterexample.
More generally, in any dihedral group
for , then
and both are subnormal whenever . ∎
However, we do observe the competing observation that the group generated by and is the same as the group generated by and , namely . Indeed in any group with subgroups and , so the condition of permutability is one which must be tested as complexes (sets ), not as subgroups. This is a consequence of the following general result:
if and only if .
We will show that every element in can be written in the form for some and . To see this first note every element in is a word over elements in and in . If the word involves only elements in or only elements in then we are done. Now for induction suppose all words of length in can be expressed in the form for some and . Then given a word of length we have either for and a word of length , in which case we are done, or for some and a word of length . Then by induction form some and . Hence . Then so there exists and such that . Thus is of the desired format. Hence so .
For the converse suppose . Then . This means for every and there exists and such that . Thus
Thus and . ∎
This helps illustrate how permutability is such a useful condition in the study of subgroup lattices (one of Ore’s main research interests). For these are the subgroups whose complexes are also subgroups. Thus we can relate the order of to the order of and and many other combinatorial relations.
|Title||example of non-permutable subgroup|
|Date of creation||2013-03-22 16:15:56|
|Last modified on||2013-03-22 16:15:56|
|Last modified by||Algeboy (12884)|