Iwasawa group

In mathematics a group is sometimes called an Iwasawa group or M-group or modular group if its lattice of subgroups is modular.

Finite modular groups are also called Iwasawa groups, after harv|Iwasawa|1941 where they were classified. Both finite and infinite M-groups are presented in textbook form in harv|Schmidt|1994|loc=Ch. 2.4. Modern study includes harv|Zimmerman|1989. A finite "p"-group is a modular group if and only if every subgroup is permutable, by harv|Schmidt|1994|loc=Lemma 2.3.2, p. 55. 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.

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