Normal subgroup

In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup)[1] is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup of the group is normal in if and only if for all and The usual notation for this relation is

Normal subgroups are important because they (and only they) can be used to construct quotient groups of the given group. Furthermore, the normal subgroups of are precisely the kernels of group homomorphisms with domain which means that they can be used to internally classify those homomorphisms.

Évariste Galois was the first to realize the importance of the existence of normal subgroups.[2]

Definitions

A subgroup of a group is called a normal subgroup of if it is invariant under conjugation; that is, the conjugation of an element of by an element of is always in [3] The usual notation for this relation is

Equivalent conditions

For any subgroup of the following conditions are equivalent to being a normal subgroup of Therefore, any one of them may be taken as the definition:

  • The image of conjugation of by any element of is a subset of [4]
  • The image of conjugation of by any element of is equal to [4]
  • For all the left and right cosets and are equal.[4]
  • The sets of left and right cosets of in coincide.[4]
  • The product of an element of the left coset of with respect to and an element of the left coset of with respect to is an element of the left coset of with respect to : for all if and then
  • is a union of conjugacy classes of [2]
  • is preserved by the inner automorphisms of [5]
  • There is some group homomorphism whose kernel is [2]
  • For all and the commutator is in
  • Any two elements commute regarding the normal subgroup membership relation: for all if and only if

Examples

For any group the trivial subgroup consisting of just the identity element of is always a normal subgroup of Likewise, itself is always a normal subgroup of (If these are the only normal subgroups, then is said to be simple.)[6] Other named normal subgroups of an arbitrary group include the center of the group (the set of elements that commute with all other elements) and the commutator subgroup [7][8] More generally, since conjugation is an isomorphism, any characteristic subgroup is a normal subgroup.[9]

If is an abelian group then every subgroup of is normal, because A group that is not abelian but for which every subgroup is normal is called a Hamiltonian group.[10]

A concrete example of a normal subgroup is the subgroup of the symmetric group consisting of the identity and both three-cycles. In particular, one can check that every coset of is either equal to itself or is equal to On the other hand, the subgroup is not normal in since [11]

In the Rubik's Cube group, the subgroups consisting of operations which only affect the orientations of either the corner pieces or the edge pieces are normal.[12]

The translation group is a normal subgroup of the Euclidean group in any dimension.[13] This means: applying a rigid transformation, followed by a translation and then the inverse rigid transformation, has the same effect as a single translation. By contrast, the subgroup of all rotations about the origin is not a normal subgroup of the Euclidean group, as long as the dimension is at least 2: first translating, then rotating about the origin, and then translating back will typically not fix the origin and will therefore not have the same effect as a single rotation about the origin.

Properties

  • If is a normal subgroup of and is a subgroup of containing then is a normal subgroup of [14]
  • A normal subgroup of a normal subgroup of a group need not be normal in the group. That is, normality is not a transitive relation. The smallest group exhibiting this phenomenon is the dihedral group of order 8.[15] However, a characteristic subgroup of a normal subgroup is normal.[16] A group in which normality is transitive is called a T-group.[17]
  • The two groups and are normal subgroups of their direct product
  • If the group is a semidirect product then is normal in though need not be normal in
  • Normality is preserved under surjective homomorphisms;[18] that is, if is a surjective group homomorphism and is normal in then the image is normal in
  • Normality is preserved by taking inverse images;[18] that is, if is a group homomorphism and is normal in then the inverse image is normal in
  • Normality is preserved on taking direct products;[19] that is, if and then
  • Every subgroup of index 2 is normal. More generally, a subgroup, of finite index, in contains a subgroup, normal in and of index dividing called the normal core. In particular, if is the smallest prime dividing the order of then every subgroup of index is normal.[20]
  • The fact that normal subgroups of are precisely the kernels of group homomorphisms defined on accounts for some of the importance of normal subgroups; they are a way to internally classify all homomorphisms defined on a group. For example, a non-identity finite group is simple if and only if it is isomorphic to all of its non-identity homomorphic images,[21] a finite group is perfect if and only if it has no normal subgroups of prime index, and a group is imperfect if and only if the derived subgroup is not supplemented by any proper normal subgroup.

Lattice of normal subgroups

Given two normal subgroups, and of their intersection and their product are also normal subgroups of

The normal subgroups of form a lattice under subset inclusion with least element, and greatest element, The meet of two normal subgroups, and in this lattice is their intersection and the join is their product.

The lattice is complete and modular.[19]

Normal subgroups, quotient groups and homomorphisms

If is a normal subgroup, we can define a multiplication on cosets as follows:

This relation defines a mapping To show that this mapping is well-defined, one needs to prove that the choice of representative elements does not affect the result. To this end, consider some other representative elements Then there are such that It follows that

where we also used the fact that is a normal subgroup, and therefore there is such that This proves that this product is a well-defined mapping between cosets.

With this operation, the set of cosets is itself a group, called the quotient group and denoted with There is a natural homomorphism, given by This homomorphism maps into the identity element of which is the coset [22] that is,

In general, a group homomorphism, sends subgroups of to subgroups of Also, the preimage of any subgroup of is a subgroup of We call the preimage of the trivial group in the kernel of the homomorphism and denote it by As it turns out, the kernel is always normal and the image of is always isomorphic to (the first isomorphism theorem).[23] In fact, this correspondence is a bijection between the set of all quotient groups of and the set of all homomorphic images of (up to isomorphism).[24] It is also easy to see that the kernel of the quotient map, is itself, so the normal subgroups are precisely the kernels of homomorphisms with domain [25]

See also

Operations taking subgroups to subgroups

  • Normalizer
  • Conjugate closure
  • Normal core

Subgroup properties complementary (or opposite) to normality

  • Malnormal subgroup
  • Contranormal subgroup
  • Abnormal subgroup
  • Self-normalizing subgroup

Subgroup properties stronger than normality

  • Characteristic subgroup
  • Fully characteristic subgroup

Subgroup properties weaker than normality

  • Subnormal subgroup
  • Ascendant subgroup
  • Descendant subgroup
  • Quasinormal subgroup
  • Seminormal subgroup
  • Conjugate permutable subgroup
  • Modular subgroup
  • Pronormal subgroup
  • Paranormal subgroup
  • Polynormal subgroup
  • C-normal subgroup

Notes

  1. Bradley 2010, p. 12.
  2. Cantrell 2000, p. 160.
  3. Dummit & Foote 2004.
  4. Hungerford 2003, p. 41.
  5. Fraleigh 2003, p. 141.
  6. Robinson 1996, p. 16.
  7. Hungerford 2003, p. 45.
  8. Hall 1999, p. 138.
  9. Hall 1999, p. 32.
  10. Hall 1999, p. 190.
  11. Judson 2020, Section 10.1.
  12. Bergvall et al. 2010, p. 96.
  13. Thurston 1997, p. 218.
  14. Hungerford 2003, p. 42.
  15. Robinson 1996, p. 17.
  16. Robinson 1996, p. 28.
  17. Robinson 1996, p. 402.
  18. Hall 1999, p. 29.
  19. Hungerford 2003, p. 46.
  20. Robinson 1996, p. 36.
  21. Dõmõsi & Nehaniv 2004, p. 7.
  22. Hungerford 2003, pp. 42–43.
  23. Hungerford 2003, p. 44.
  24. Robinson 1996, p. 20.
  25. Hall 1999, p. 27.

References

  • Bergvall, Olof; Hynning, Elin; Hedberg, Mikael; Mickelin, Joel; Masawe, Patrick (16 May 2010). "On Rubik's Cube" (PDF). KTH. Cite journal requires |journal= (help)
  • Cantrell, C.D. (2000). Modern Mathematical Methods for Physicists and Engineers. Cambridge University Press. ISBN 978-0-521-59180-5.
  • Dõmõsi, Pál; Nehaniv, Chrystopher L. (2004). Algebraic Theory of Automata Networks. SIAM Monographs on Discrete Mathematics and Applications. SIAM.
  • Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9.
  • Fraleigh, John B. (2003). A First Course in Abstract Algebra (7th ed.). Addison-Wesley. ISBN 978-0-321-15608-2.
  • Hall, Marshall (1999). The Theory of Groups. Providence: Chelsea Publishing. ISBN 978-0-8218-1967-8.
  • Hungerford, Thomas (2003). Algebra. Graduate Texts in Mathematics. Springer.
  • Judson, Thomas W. (2020). Abstract Algebra: Theory and Applications.
  • Robinson, Derek J. S. (1996). A Course in the Theory of Groups. Graduate Texts in Mathematics. 80 (2nd ed.). Springer-Verlag. ISBN 978-1-4612-6443-9. Zbl 0836.20001.
  • Thurston, William (1997). Levy, Silvio (ed.). Three-dimensional geometry and topology, Vol. 1. Princeton Mathematical Series. Princeton University Press. ISBN 978-0-691-08304-9.
  • Bradley, C. J. (2010). The mathematical theory of symmetry in solids : representation theory for point groups and space groups. Oxford New York: Clarendon Press. ISBN 978-0-19-958258-7. OCLC 859155300.

Further reading

  • I. N. Herstein, Topics in algebra. Second edition. Xerox College Publishing, Lexington, Mass.-Toronto, Ont., 1975. xi+388 pp.
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