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: Memetics.

gollark: I'm not sure what percentile the chimps are at.

gollark: It's something about remembering the order of some squares.

gollark: Observe, highly accurate and good random internet tests.

gollark: Apparently verbal and whatever-else memory are generally quite correlated, so this is weird.

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

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

Iwasawa group

Examples

See also

Further reading

References