Iwasawa group

In mathematics, a group is called an Iwasawa group, M-group or modular group if its lattice of subgroups is modular. Alternatively, a group "G" is called an Iwasawa group when every subgroup of "G" is permutable in "G" .
 proved that a "p"-group "G" is an Iwasawa group if and only if one of the following cases happens:
In , Iwasawa's proof was deemed to have essential gaps, which were filled by Franco Napolitani and Zvonimir Janko. has provided an alternative proof along different lines in his textbook. As part of Schmidt's proof, he proves that a finite "p"-group is a modular group if and only if every subgroup is permutable, by .
Every subgroup of a finite "p"-group is subnormal, and those finite groups in which subnormality and permutability coincide are called PT-groups. In other words, a finite "p"-group is an Iwasawa group if and only if it is a PT-group.
Examples.
The Iwasawa group of order 16 is isomorphic to the modular group of order 16.
Further reading.
Both finite and infinite M-groups are presented in textbook form in . Modern study includes .