Normal subgroup

Summary

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]

DefinitionsEdit

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 conditionsEdit

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  [citation needed]
  • Any two elements commute regarding the normal subgroup membership relation: for all     if and only if  [citation needed]

ExamplesEdit

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] This illustrates the general fact that any subgroup   of index two is normal.

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.

PropertiesEdit

  • 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 subgroupsEdit

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 homomorphismsEdit

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 alsoEdit

NotesEdit

  1. ^ Bradley 2010, p. 12.
  2. ^ a b c Cantrell 2000, p. 160.
  3. ^ Dummit & Foote 2004.
  4. ^ a b c d 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. ^ a b Hall 1999, p. 29.
  19. ^ a b 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.

ReferencesEdit

  • Bergvall, Olof; Hynning, Elin; Hedberg, Mikael; Mickelin, Joel; Masawe, Patrick (16 May 2010). "On Rubik's Cube" (PDF). KTH. {{cite journal}}: 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. Vol. 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 readingEdit

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

External linksEdit

  • Weisstein, Eric W. "normal subgroup". MathWorld.
  • Normal subgroup in Springer's Encyclopedia of Mathematics
  • Robert Ash: Group Fundamentals in Abstract Algebra. The Basic Graduate Year
  • Timothy Gowers, Normal subgroups and quotient groups
  • John Baez, What's a Normal Subgroup?