Iwasawa group¶

In mathematics, a group is called an Iwasawa group, M-group or modular group if its lattice of subgroups is modular.

Version: v1 · 2026-09-28 · History
Domain-specific #: 10167
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Group Theory → Mathematics

Core Idea¶

Iwasawa group is treated here as the recurring mathematicslogicstatistics identity summarized by this source-grounded definition: In mathematics, a group is called an Iwasawa group, M-group or modular group if its lattice of subgroups is modular. 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. The name is derived from Kenkichi Iwasawa. proved that a p-group G is an Iwasawa group if and only if one of.

Scope of Application¶

G is a Dedekind group, or. In , Iwasawa's proof was deemed to have essential gaps, which were filled by Franco Napolitani and Zvonimir Janko. has provided an alternative proof along different lines in his textbook.
G is a Dedekind group, or. 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.
G is a Dedekind group, or. Every subgroup of a finite p-group is subnormal, and those finite groups in which subnormality and permutability coincide are called PT-groups.
G is a Dedekind group, or. In other words, a finite p-group is an Iwasawa group if and only if it is a PT-group.
Examples. The Iwasawa group of order 16 is isomorphic to the modular maximal-cyclic group of order 16.

Clarity¶

A clear use of Iwasawa group names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In mathematics, a group is called an Iwasawa group, M-group or modular group if its lattice of subgroups is modular.

Manages Complexity¶

Iwasawa group compresses multiple mathematicslogicstatistics details into a stable diagnostic relation. The source shows both the central mechanism—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 .—and the practical consequence—g contains an abelian normal subgroup N such that the quotient group G/N is a cyclic group and if q denotes a generator.

Abstract Reasoning¶

Type the carrier. Identify the mathematicslogicstatistics entities to which the claim applies.
State the relation. Use the source-grounded identity: In mathematics, a group is called an Iwasawa group, M-group or modular group if its lattice of subgroups is modular.
Check operation and conditions. Every subgroup of a finite p-group is subnormal, and those finite groups in which subnormality and permutability coincide are called PT-groups.
Demand recognition evidence.

Knowledge Transfer¶

Within the home domain. Knowledge about Iwasawa group transfers literally when a new case preserves the same carrier type, relation, and recognition test. In , Iwasawa's proof was deemed to have essential gaps, which were filled by Franco Napolitani and Zvonimir Janko. 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. Beyond the home domain. No canonical parent is asserted for Iwasawa group.

Relationships to Other Abstractions¶

Current abstraction Iwasawa group Domain-specific

Parents (1) — more general patterns this builds on

Iwasawa group is a kind of Group Prime

An Iwasawa group is a group distinguished by modularity of its subgroup lattice.

Neighborhood in Abstraction Space¶

Iwasawa group sits in a moderately populated region (55^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Number Theory & Packing Conjectures (5 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08