 | This article may be
confusing or unclear to readers. (April 2015) |
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 ( 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 = n1+ps where s ≥ 1 in general, but s ≥ 2 for p=2.
In Berkovich & Janko (2008, p. 257), Iwasawa's proof was deemed to have 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.[ citation needed]
Examples
The Iwasawa group of order 16 is isomorphic to the modular maximal-cyclic group of order 16.[ citation needed]
See also
Further reading
Both finite and infinite M-groups are presented in textbook form in Schmidt (1994, Ch. 2). Modern study includes Zimmermann (1989).
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, doi: 10.4099/jjm1924.18.0_709, MR 0015118
- Schmidt, Roland (1994), Subgroup Lattices of Groups, Expositions in Math, vol. 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, S2CID 121609694
- 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, vol. 2, Walter de Gruyter, ISBN 978-3-11-020823-8
 | This group theory-related article is a stub. You can help Wikipedia by expanding it. |
| This article may be
confusing or unclear to readers. (April 2015) |
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 ( 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 = n1+ps where s ≥ 1 in general, but s ≥ 2 for p=2.
In Berkovich & Janko (2008, p. 257), Iwasawa's proof was deemed to have 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.[ citation needed]
Examples
The Iwasawa group of order 16 is isomorphic to the modular maximal-cyclic group of order 16.[ citation needed]
See also
Further reading
Both finite and infinite M-groups are presented in textbook form in Schmidt (1994, Ch. 2). Modern study includes Zimmermann (1989).
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, doi: 10.4099/jjm1924.18.0_709, MR 0015118
- Schmidt, Roland (1994), Subgroup Lattices of Groups, Expositions in Math, vol. 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, S2CID 121609694
- 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, vol. 2, Walter de Gruyter, ISBN 978-3-11-020823-8
|
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