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.

Alternatively, a group G is called an Iwasawa group when every subgroup of G is permutable in G (Ballester-Bolinches, Esteban-Romero & Asaad 2010, pp. 24–25).

Kenkichi Iwasawa (1941) proved that a p-group G is an Iwasawa group if and only if one of the following cases happens:

G is a Dedekind group, or
G contains an abelian normal subgroup N such that the quotient group G/N is a cyclic group and if q denotes a generator of G/N, then for all n ∈ N, q⁻¹nq = n^{1+p^s} where s ≥ 1 in general, but s ≥ 2 for p=2.

In Berkovich & Janko (2008, p. 257), Iwasawa's proof was deemed to have some 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).

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

gollark: Not *exactly*. There is a nonzero chance that you somehow completely failed to notice that it had an 8 on it and it lands on that, or that the dice is somehow swapped out for one with 8s on it as you roll it, or that sort of thing.

gollark: But I want them to implement self replicating spreadsheet cells to run some weird cellular automaton to run a Turing machine to parse HTML.

gollark: I don't know if it has HTTP capability, but it could totally sort of do HTML/CSS if it is.

gollark: Excel is Turing-complete isn't it?

gollark: I don't think people do much of the time, though.

References

Iwasawa, Kenkichi (1941), "Über die endlichen Gruppen und die Verbände ihrer Untergruppen", J. Fac. Sci. Imp. Univ. Tokyo. Sect. I., 4: 171–199, MR 0005721
Iwasawa, Kenkichi (1943), "On the structure of infinite M-groups", Japanese Journal of Mathematics, 18: 709–728, MR 0015118
Schmidt, Roland (1994), Subgroup Lattices of Groups, Expositions in Math, 14, Walter de Gruyter, doi:10.1515/9783110868647, ISBN 978-3-11-011213-9, MR 1292462
Zimmermann, Irene (1989), "Submodular subgroups in finite groups", Mathematische Zeitschrift, 202 (4): 545–557, doi:10.1007/BF01221589, MR 1022820
Ballester-Bolinches, Adolfo; Esteban-Romero, Ramon; Asaad, Mohamed (2010), Products of Finite Groups, Walter de Gruyter, pp. 24–25, ISBN 978-3-11-022061-2
Berkovich, Yakov; Janko, Zvonimir (2008), Groups of Prime Power Order, 2, Walter de Gruyter, ISBN 978-3-11-020823-8

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Iwasawa group

Examples

See also

Further reading

References