Kenkichi Iwasawa (1941) proved that a p-group G is an Iwasawa group if and only if one of the following cases happens: In Berkovich & Janko (2008, p. 257), Iwasawa's proof was deemed to have essential gaps, which were filled by Franco Napolitani and Zvonimir Janko.
Roland Schmidt (1994) 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 (Schmidt 1994, Lemma 2.3.2, p. 55).
In other words, a finite p-group is an Iwasawa group if and only if it is a PT-group.
[citation needed] Both finite and infinite M-groups are presented in textbook form in Schmidt (1994, Ch.