Normal subgroup

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In abstract algebra, a normal 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 H of a group G is normal in G if and only if gH = Hg for all g in G. The definition of normal subgroup implies that the sets of left and right cosets coincide. In fact, a seemingly weaker condition that the sets of left and right cosets coincide also implies that the subgroup H of a group G is normal in G. Normal subgroups (and only normal subgroups) can be used to construct quotient groups from a given group.

Évariste Galois was the first to realize the importance of the existence of normal subgroups.[1]

Definitions[edit]

A subgroup N of a group G is called a normal subgroup if it is invariant under conjugation; that is, the conjugation of an element of N by an element of G is always in N:[2]

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

  • Any two elements commute regarding the normal subgroup membership relation: g, hG, ghNhgN.
  • The image of conjugation of N by any element of G is a subset of N: gG, gNg−1N.[3]
  • The image of conjugation of N by any element of G is N: gG, gNg−1 = N.[3]
  • gG, gN = Ng.[3]
  • The sets of left and right cosets of N in G coincide.[3]
  • The product of an element of the left coset of N with respect to g and an element of the left coset of N with respect to h is an element of the left coset of N with respect to gh: x, y, g, hG, xgN and yhNxy(gh)N.
  • N is a union of conjugacy classes of G: N = ⋃gN Cl(g).[1]
  • N is preserved by inner automorphisms.[4]
  • There is some homomorphism on G for which N is the kernel: ∃φ ∈ Hom(G) ∣ ker φ = N.[1]

The last condition 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,[5] 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.

Examples[edit]

Properties[edit]

  • Normality is preserved upon surjective homomorphisms, and is also preserved upon taking inverse images.[13]
  • Normality is preserved on taking direct products.[14]
  • If H is a normal subgroup of G, and K is a subgroup of G containing H, then H is a normal subgroup of K.[15]
  • 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.[16] However, a characteristic subgroup of a normal subgroup is normal.[17] A group in which normality is transitive is called a T-group.[18]
  • Every subgroup of index 2 is normal. More generally, a subgroup, H, of finite index, n, in G contains a subgroup, K, normal in G and of index dividing n! called the normal core. In particular, if p is the smallest prime dividing the order of G, then every subgroup of index p is normal.[19]

Lattice of normal subgroups[edit]

The normal subgroups of a group, G, form a lattice under subset inclusion with least element, {e} , and greatest element, G. Given two normal subgroups, N and M, in G, meet is defined as

and join is defined as

The lattice is complete and modular.[14]

Normal subgroups and homomorphisms[edit]

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

With this operation, the set of cosets is itself a group, called the quotient group and denoted G/N. There is a natural homomorphism, f: GG/N given by f(a) = aN. The image f(N) consists only of the identity element of G/N, the coset eN = N.[20]

In general, a group homomorphism, f: GH sends subgroups of G to subgroups of H. Also, the preimage of any subgroup of H is a subgroup of G. We call the preimage of the trivial group {e} in H the kernel of the homomorphism and denote it by ker(f). As it turns out, the kernel is always normal and the image of G, f(G), is always isomorphic to G/ker(f) (the first isomorphism theorem).[21] In fact, this correspondence is a bijection between the set of all quotient groups of G, G/N, and the set of all homomorphic images of G (up to isomorphism).[22] It is also easy to see that the kernel of the quotient map, f: GG/N, is N itself, so the normal subgroups are precisely the kernels of homomorphisms with domain G.[23]

See also[edit]

Notes[edit]

  1. ^ a b c Cantrell 2000, p. 160.
  2. ^ Dummit & Foote 2004.
  3. ^ a b c d Hungerford 2003, p. 41.
  4. ^ Fraleigh 2003, p. 141.
  5. ^ Dõmõsi & Nehaniv 2004, p. 7.
  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. ^ Thurston 1997, p. 218.
  12. ^ Bergvall et al. 2010, p. 96.
  13. ^ Hall 1999, p. 29.
  14. ^ a b Hungerford 2003, p. 46.
  15. ^ Hungerford 2003, p. 42.
  16. ^ Robinson 1996, p. 17.
  17. ^ Robinson 1996, p. 28.
  18. ^ Robinson 1996, p. 402.
  19. ^ Robinson 1996, p. 36.
  20. ^ Hungerford 2003, pp. 42–43.
  21. ^ Hungerford 2003, p. 44.
  22. ^ Robinson 1996, p. 20.
  23. ^ Hall 1999, p. 27.

References[edit]

  • Bergvall, Olof; Hynning, Elin; Hedberg, Mikael; Mickelin, Joel; Masawe, Patrick (16 May 2010). "On Rubik's Cube" (PDF). KTH.
  • 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.
  • 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.

Further reading[edit]

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

External links[edit]