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In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
is called an
Iwasawa group, M-group or modular group if its
lattice of subgroups
In mathematics, the lattice of subgroups of a group G is the lattice whose elements are the subgroups of G, with the partial ordering being set inclusion.
In this lattice, the join of two subgroups is the subgroup generated by their union, ...
is
modular. Alternatively, a group ''G'' is called an Iwasawa group when every subgroup of ''G'' is
permutable in ''G'' .
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
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...
''N'' such that the
quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored out"). For ex ...
''G/N'' is a
cyclic group
In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...
and if ''q'' denotes a generator of ''G/N'', then for all ''n'' ∈ ''N'', ''q''
−1''nq'' = ''n''
1+''p''''s'' where ''s'' ≥ 1 in general, but ''s'' ≥ 2 for ''p''=2.
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 .
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
The Iwasawa group of order 16 is isomorphic to the
modular maximal-cyclic group of order 16.
See also
*
Modular law for groups
Further reading
Both finite and infinite M-groups are presented in textbook form in . Modern study includes .
References
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*
Finite groups
Properties of groups
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