Difference between revisions of "Normality MA453S11Walther" - Rhea

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**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)

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)

Factor Groups and Kernels of Homomorphisms: 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)

Normal subgroups are also important since they are the kernels of homomorphisms on the group.

Links to pages on normal subgroups:

- http://mathworld.wolfram.com/NormalSubgroup.html

- http://eom.springer.de/N/n067690.htm

- http://math.ucr.edu/home/baez/normal.html

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