Normality
A student project for MA453: "Abstract Algebra"
**Reference citations are denoted throughout as (reference #) after the cited information** - Mark Knight
Preliminary Definitions
Let $ G $ be a group and $ N $ be a subgroup of $ G $.
The element $ gng^{-1} $ is called the conjugate of $ n\in N $ by $ g $.
The set $ gNg^{-1} =\{ {gng^{-1} | n\in N}\} $ is called the conjugate of $ N $ by $ g $.
The element $ g $ normalizes $ N $ if $ gNg^{-1} = N $.
A subgroup $ N $ of a group $ G $ is said to be normal if every element of $ G $ normalizes $ N $. That is, if $ gNg^{-1} = N $ for all g in G. (reference #2)
Equivalent definitions of Normality
Let $ G $ be a group and $ N $ be a subgroup of $ G $. The following are equivalent:
1. $ gNg^{-1}\subseteq N $ for all $ g\in G $.
2. $ gNg^{-1} = N $ for all $ g\in G $.
3. $ gN = Ng $ for all $ g\in G $. That is, the left and right cosets are equal. (reference #1)
4. $ N $ is the kernel of some homomorphism on $ G $. (reference #2)
The equivalence of (1), (2) and (3) above is proved here:
Lemma: If $ N \le G $ then $ (aN)(bN) = abN $ for all $ a,b \in G $ $ \Leftrightarrow $ $ gNg^{-1} = N $ for all $ g \in G $.
For $ \Leftarrow $ we have then $ (aN)(bN) = a(Nb)N = abNN = abN $.
For $ \Rightarrow $ then $ gNg^{-1} \subseteq gNg^{-1}N $ since $ 1\in N $ and by the hypothesis $ (gN)(g^{-1}N) = gg^{-1}N (=N) $. Then we have $ gNg^{-1} \subseteq N $ which implies that $ N\subseteq g^{-1}Ng $. Because this result holds for all $ g \in G $, we have $ N \subseteq gNg^{-1} $ and the desired result follows. $ \Box $ (reference #1)
Examples of Normal Subgroups
1. Every subgroup of an Abelian group is normal because for elements a in G and h in N, ah = ha. (reference #3)
2. The trivial subgroup consisting only of the identity is normal, as is the entire group itself. (refernce #4). If it is the case that {1} and {G} are the only normal subgroups of G, then G is said to be simple. (reference #2)
3. The center of a group is normal because, again, ah = ha for a in G and h in Z(G). (reference #3)
4. The subgroup of rotations in the dihedral groups are normal in the dihedral groups. (reference #3) An explicit example of this can be shown with the group D3.
5. SL (n,R) is normal in GL (n,R) because if A is a nonsingular n by n matrix and B is n by n with determinant 1, then det$ ABA^{-1} $ = $ detAdetBdetA^{-1} $ = detB = 1. (reference #1)
Further examples can be found in the links.
Factor Groups, Kernels of Homomorphisms and Galois Extensions: The Significance of Normal Subgroups
When a subgroup N of a group G is normal, then the set of cosets of N in G is called the factor group of G by N. If G is a group and N is a normal subgroup of G, then the set {aN | a $ \in $ G} is a group under the operation (aN)(bN) = abN. It is often possible to tell information about a larger group by studying one of its factor groups. (reference #3)
An example in Gallian shows how the factor group Z/4Z can be constructed from Z and 4Z. First the left cosets of 4Z in Z are determined. These are 0 + 4Z = {..., -8, -4, 0, 4, 8,...}; 1 + 4Z = (1,5,9,...; -3,-7,-11,...}; 2 + 4Z = {2,6,10,...; -2, -6, -10,...}; and 3 + 4Z = {3,7,11,...; -1,-5,-9,...}. The structure of the group is determined by Cayley table and is shown to be isomorphic to {0,1,2,3} under addition mod 4. (reference #3)
Normal subgroups are also important since they are the kernels of homomorphisms on the group G. For a homomorphism p: G $ \rightarrow $ H, then the image of p is isomorphic to G/ker(p). This is the first isomorphism theorem. (reference #1)
If the fields E/K, E/F and K/F are Galois field extensions, then there is a one-to-one correspondence between the normal subgroups of Gal(E,F) and the Galois extension E containing K containing F. The associated normal subgroup of Gal(E/F) is the elements of Gal(E/F) that don't move K. This fact can be used to show that the general degree-5 polynomial is not solvable by radicals. (reference #4)
Theorems of Normal Subgroups
These three theorems show how information from a factor group of G implies information about G itself.
Theorem: If G is a group with center Z(G) then if G/Z(G) is cyclic then G is Abelian. (reference #3)
Theorem: For any group G, G/Z(G) is isomorphic to Inn(G) (where "Inn" denotes the inner automorphisms of the group G) (reference #3)
Theorem: Let G be a finite Abelian group and let p be a prime that divides the order of G. Then G has an element of order p. (reference #3)
Links to interesting pages on normal subgroups:
- http://groupprops.subwiki.org/wiki/Normal_subgroup
- http://mathworld.wolfram.com/NormalSubgroup.html
- http://eom.springer.de/N/n067690.htm
- http://math.ucr.edu/home/baez/normal.html (This link contains an interesting geometric interpretation of normal subgroups.)
- http://marauder.millersville.edu/~bikenaga/abstractalgebra/normal/normal.html
- http://www.jstor.org/stable/2690280?seq=1
- http://www.youtube.com/watch?v=NwRLh-bbvts
References:
(1) http://www.math.uiuc.edu/~r-ash/Algebra/Chapter1.pdf
(2) Dummit, D.S. & Foote, R.M. (1991). Abstract Algebra. United States: Prentice Hall.
(3) Gallian, J.A. (2010). Contemporary Abstract Algebra. United States: Brooks/Cole.
(4) MA 453 lecture notes, Professor Uli Walther
