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.
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.
4. $ N $ is the kernel of some homomorphism on $ G $.
The equivalence of (1), (2) and (3) above is proved below:
Lemma: If $ N \le G $ then $ (aN)(bN) = abN $ for all $ a,b \in G $ $ \Leftrightarrow $ $ gNg^{-1} = N $ for all $ g \in G $.
Links to pages on normal subgroups:
(1) http://mathworld.wolfram.com/NormalSubgroup.html
(2) http://eom.springer.de/N/n067690.htm
(3) http://math.ucr.edu/home/baez/normal.html
References:
- http://www.math.uiuc.edu/~r-ash/Algebra/Chapter1.pdf
- Dummit, D.S. & Foote, R.M. (1991). Abstract Algebra. United States: Prentice Hall.
- Gallian, J.A. (2010). Contemporary Abstract Algebra. United States: Brooks/Cole.
- MA 453 lecture notes, Professor Uli Walther
