example of non-permutable subgroup

Example 1.

There are groups (even finitely generatedMathworldPlanetmathPlanetmathPlanetmath) with subnormal subgroupsMathworldPlanetmath which are not permutable.


Let D8 be the dihedral groupMathworldPlanetmath of order 8. The classic presentationMathworldPlanetmathPlanetmathPlanetmath is


As the group is nilpotentPlanetmathPlanetmath we know every subgroupMathworldPlanetmathPlanetmath is subnormal; however, not every subgroup is permutable. In particular, observe that for two general subgroups H and K of D8, it may be possible that HK is not a subgroup. In this situation we find our counterexample.




More generally, in any dihedral group


for n>2, then


and both are subnormal whenever n=2i. ∎

However, we do observe the competing observation that the group generated by b and ab is the same as the group generated by ab and b, namely D8. Indeed in any group with subgroups H and K, H,K=K,H so the condition of permutability is one which must be tested as complexes (sets HK), not as subgroups. This is a consequence of the following general result:

Claim 1.

HK=KH if and only if HK=H,K.


We will show that every element in H,K can be written in the form hk for some hH and kK. To see this first note every element in H,K is a word over elements in H and in K. If the word involves only elements in H or only elements in K then we are done. Now for inductionMathworldPlanetmath suppose all words of length m in H,K can be expressed in the form hk for some hH and kK. Then given a word of length m+1 we have either hw for hH and w a word of length m, in which case we are done, or kw for some kK and w a word of length m. Then by induction w=hk form some hH and kK. Hence kw=khk. Then khKH=HK so there exists hH and k′′K such that kh=hk′′. Thus kw=hk′′k=h(k′′k) is of the desired format. Hence HKH,KHK so HK=H,K.

For the converseMathworldPlanetmath suppose HK=H,K. Then KHHK. This means for every kK and hH there exists hH and kK such that k-1h-1=hk. Thus


Thus HKKH and KH=HK. ∎

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 H,K to the order of H and K and many other combinatorial relationsMathworldPlanetmath.

Title example of non-permutable subgroup
Canonical name ExampleOfNonpermutableSubgroup
Date of creation 2013-03-22 16:15:56
Last modified on 2013-03-22 16:15:56
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 10
Author Algeboy (12884)
Entry type Example
Classification msc 20E07