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- Q6100707 subject Q7003933.
- Q6100707 subject Q7029099.
- Q6100707 abstract "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 et al. 2010, pp. 24-25).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 some essential gaps, which were filled by F. Napolitani and Z. Janko. 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.".
- Q6100707 wikiPageWikiLink Q1138961.
- Q6100707 wikiPageWikiLink Q1538614.
- Q6100707 wikiPageWikiLink Q1573561.
- Q6100707 wikiPageWikiLink Q181296.
- Q6100707 wikiPageWikiLink Q231454.
- Q6100707 wikiPageWikiLink Q2361983.
- Q6100707 wikiPageWikiLink Q245462.
- Q6100707 wikiPageWikiLink Q286972.
- Q6100707 wikiPageWikiLink Q338585.
- Q6100707 wikiPageWikiLink Q3538331.
- Q6100707 wikiPageWikiLink Q395.
- Q6100707 wikiPageWikiLink Q7003933.
- Q6100707 wikiPageWikiLink Q7029099.
- Q6100707 wikiPageWikiLink Q7269511.
- Q6100707 wikiPageWikiLink Q743179.
- Q6100707 wikiPageWikiLink Q83478.
- Q6100707 wikiPageWikiLink Q847858.
- Q6100707 wikiPageWikiLink Q933692.
- Q6100707 comment "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 et al. 2010, pp.".
- Q6100707 label "Iwasawa group".