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

Lattice of subgroups

In mathematics, the lattice of subgroups of a group G is the lattice whose elements are the subgroups of G, with the partial order relation being set inclusion....

 is modular

Modular lattice

In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self-dual condition:Modular law: x ≤ b implies x ∨  =  ∧ b,where  ≤  is the partial order, and  ∨  and  ∧  are...

.

Finite modular groups are also called Iwasawa groups, after where they were classified. Both finite and infinite M-groups are presented in textbook form in . Modern study includes . A finite p-group

P-group

In mathematics, given a prime number p, a p-group is a periodic group in which each element has a power of p as its order: each element is of prime power order. That is, for each element g of the group, there exists a nonnegative integer n such that g to the power pn is equal to the identity element...

 is a modular group if and only if every subgroup is permutable, by . Every subgroup of a finite p-group is subnormal

Subnormal subgroup

In mathematics, in the field of group theory, a subgroup H of a given group G is a subnormal subgroup of G if there is a finite chain of subgroups of the group, each one normal in the next, beginning at H and ending at G....

, 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.