Iwasawa group

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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 (Iwasawa 1941) where they were classified. Both finite and infinite M-groups are presented in textbook form in (Schmidt 1994, Ch. 2.4). Modern study includes (Zimmerman 1989). 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). 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.

[edit] References

• Iwasawa, Kenkiti (1941), “Über die endlichen Gruppen und die Verbände ihrer Untergruppen”, J. Fac. Sci. Imp. Univ. Tokyo. Sect. I. 4: 171–199, MR0005721 
• Iwasawa, Kenkiti (1943), “On the structure of infinite M-groups”, Jap. J. Math. 18: 709–728, MR0015118 
• Schmidt, Roland (1994), Subgroup Lattices of Groups, vol. 14, Expositions in Math, Walter de Gruyter, MR1292462, ISBN 978-3-11-011213-9 
• Zimmermann, Irene (1989), “Submodular subgroups in finite groups”, Mathematische Zeitschrift 202 (4): 545–557, MR1022820, ISSN 0025-5874, DOI 10.1007/BF01221589