Lattice of subgroups
In mathematics, the lattice of subgroups of a group is the lattice whose elements are the subgroups of , with the
Example
The
This example also shows that the lattice of all subgroups of a group is not a
Properties
For any A, B, and C subgroups of a group with A ≤ C (A a subgroup of C) then AB ∩ C = A(B ∩ C); the multiplication here is the
The
The Zassenhaus lemma gives an isomorphism between certain combinations of quotients and products in the lattice of subgroups.
In general, there is no restriction on the shape of the lattice of subgroups, in the sense that every lattice is
Characteristic lattices
Subgroups with certain properties form lattices, but other properties do not.
- Normal subgroups always form a modular lattice. In fact, the essential property that guarantees that the lattice is modular is that subgroups commute with each other, i.e. that they are quasinormal subgroups.
- Nilpotent normal subgroups form a lattice, which is (part of) the content of Fitting's theorem.
- A class of groups is called a Fitting class if it is closed under isomorphism, subnormal subgroups, and products of subnormal subgroups. For any Fitting class F, both the subnormal F-subgroups and the normal F-subgroups form lattices. This generalizes the above with F the class of nilpotent groups, and another example is with F the class of solvable groups.
- Central subgroups form a lattice.
However, neither finite subgroups nor torsion subgroups form a lattice: for instance, the free product is generated by two
The fact that normal subgroups form a modular lattice is a particular case of a more general result, namely that in any
Characterizing groups by their subgroup lattices
Groups whose lattice of subgroups is a complemented lattice are called complemented groups (Zacher 1953), and groups whose lattice of subgroups are modular lattices are called Iwasawa groups or modular groups (Iwasawa 1941). Lattice-theoretic characterizations of this type also exist for solvable groups and perfect groups (Suzuki 1951).
References
- Aschbacher, M. (2000). Finite Group Theory. Cambridge University Press. p. 6. ISBN 978-0-521-78675-1.
- Baer, Reinhold (1939). "The significance of the system of subgroups for the structure of the group". JSTOR 2371383.
- Cohn, Paul Moritz (2000). Classic algebra. Wiley. p. 248. ISBN 978-0-471-87731-8.
- Iwasawa, Kenkiti (1941), "Über die endlichen Gruppen und die Verbände ihrer Untergruppen", J. Fac. Sci. Imp. Univ. Tokyo. Sect. I., 4: 171–199, MR 0005721
- Kearnes, Keith; Kiss, Emil W. (2013). The Shape of Congruence Lattices. American Mathematical Soc. p. 3. ISBN 978-0-8218-8323-5.
- MR 1545977.
- MR 1546048.
- Robinson, Derek (1996). A Course in the Theory of Groups. Springer Science & Business Media. p. 15. ISBN 978-0-387-94461-6.
- Rottlaender, Ada (1928). "Nachweis der Existenz nicht-isomorpher Gruppen von gleicher Situation der Untergruppen". S2CID 120596994.
- Schmidt, Roland (1994). Subgroup Lattices of Groups. Expositions in Math. Vol. 14. Walter de Gruyter. ISBN 978-3-11-011213-9. Reviewby Ralph Freese in Bull. AMS 33 (4): 487–492.
- JSTOR 1990375.
- Suzuki, Michio (1956). Structure of a Group and the Structure of its Lattice of Subgroups. Berlin: Springer Verlag.
- Yakovlev, B. V. (1974). "Conditions under which a lattice is isomorphic to a lattice of subgroups of a group". Algebra and Logic. 13 (6): 400–412. S2CID 119943975.
- Silcock, Howard L. (1977). "Generalized wreath products and the lattice of normal subgroups of a group" (PDF). Algebra Universalis. 7: 361–372.
- Zacher, Giovanni (1953). "Caratterizzazione dei gruppi risolubili d'ordine finito complementati". MR 0057878.