Iwasawa group
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In mathematics, a group is called an Iwasawa group, 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 {{harv|Ballester-Bolinches|Esteban-Romero|Asaad|2010|pp=24–25}}.
{{harvs|txt|last=Iwasawa|first=Kenkichi|Kenkichi Iwasawa|year=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 = n1+ps where s ≥ 1 in general, but s ≥ 2 for p=2.
In {{harvtxt|Berkovich|Janko|2008|p=257}}, Iwasawa's proof was deemed to have essential gaps, which were filled by Franco Napolitani and Zvonimir Janko. {{harvs|txt|last=Schmidt|first=Roland| year=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 {{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.{{citation needed|date=April 2015}}
Examples
The Iwasawa group of order 16 is isomorphic to the modular maximal-cyclic group of order 16.{{cn|date=May 2021}}
See also
Further reading
Both finite and infinite M-groups are presented in textbook form in {{harvtxt|Schmidt|1994|loc=Ch. 2}}. Modern study includes {{harvtxt|Zimmermann|1989}}.
References
- {{Citation | last=Iwasawa | first=Kenkichi| authorlink=Kenkichi Iwasawa| title=Über die endlichen Gruppen und die Verbände ihrer Untergruppen |mr=0005721 | year=1941 | journal=J. Fac. Sci. Imp. Univ. Tokyo. Sect. I. | volume=4 | pages=171–199}}
- {{Citation | last=Iwasawa | first=Kenkichi| authorlink=Kenkichi Iwasawa| title=On the structure of infinite M-groups |mr=0015118 | year=1943 | journal=Japanese Journal of Mathematics | volume=18 | pages=709–728| doi=10.4099/jjm1924.18.0_709| doi-access=free }}
- {{Citation | last1=Schmidt | first1=Roland | title=Subgroup Lattices of Groups | publisher=Walter de Gruyter | series=Expositions in Math | isbn=978-3-11-011213-9 |mr=1292462 | year=1994 | volume=14 | doi=10.1515/9783110868647}}
- {{Citation | last1=Zimmermann | first1=Irene | title=Submodular subgroups in finite groups | doi=10.1007/BF01221589 |mr=1022820 | year=1989 | journal=Mathematische Zeitschrift | volume=202 | issue=4 | pages=545–557| s2cid=121609694 }}
- {{citation|first1=Adolfo|last1=Ballester-Bolinches|first2=Ramon|last2=Esteban-Romero|first3=Mohamed|last3=Asaad|title=Products of Finite Groups|year=2010|publisher=Walter de Gruyter|isbn=978-3-11-022061-2|pages=24–25}}
- {{citation|first1=Yakov|last1=Berkovich|first2=Zvonimir|last2=Janko|title=Groups of Prime Power Order|year=2008|publisher=Walter de Gruyter|isbn=978-3-11-020823-8|volume=2}}
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