(New page: '''Links to pages on normal subgroups:''' (1) http://mathworld.wolfram.com/NormalSubgroup.html (2) http://eom.springer.de/N/n067690.htm (3) http://www.math.uiuc.edu/~r-ash/Algebra/Chapt...) |
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| + | '''Preliminary Definitions''' | ||
| + | ---- | ||
| + | Let <math>G</math> be a group and <math>N</math> be a subgroup of <math>G</math>. | ||
| + | |||
| + | The element <math>gng^{-1}</math> is called the ''conjugate'' of <math>n\in N</math> by <math>g</math>. | ||
| + | |||
| + | The set <math>gNg^{-1} =\{ {gng^{-1} | n\in N}\}</math> is called the ''conjugate of <math>N</math> by <math>g</math>.'' | ||
| + | |||
| + | The element <math>g</math> ''normalizes'' <math>N</math> if <math>gNg^{-1} = N</math>. | ||
| + | |||
| + | A subgroup <math>N</math> of a group <math>G</math> is said to be ''normal'' if every element of <math>G</math> normalizes <math>N</math>. That is, if <math>gNg^{-1} = N</math> for all g in G. | ||
| + | |||
| + | |||
| + | '''Equivalent definitions of Normality''' | ||
| + | ---- | ||
| + | Let <math>G</math> be a group and <math>N</math> be a subgroup of <math>G</math>. The following are equivalent: | ||
| + | |||
| + | 1. <math>gNg^{-1}\subseteq N</math> for all <math> g\in G</math>. | ||
| + | |||
| + | 2. <math>gNg^{-1} = N</math> for all <math> g\in G</math>. | ||
| + | |||
| + | 3. <math>gN = Ng</math> for all <math> g\in G</math>. That is, the left and right cosets are equal. | ||
| + | |||
| + | 4. <math>N</math> is the kernel of some homomorphism on <math> G </math>. | ||
| + | |||
| + | |||
| + | The equivalence of (1), (2) and (3) above is proved below: | ||
| + | |||
| + | Lemma: If <math> N \le G</math> then <math> (aN)(bN) = abN </math> for all <math> a,b \in G</math> <math> \Leftrightarrow </math> <math> gNg^{-1} = N </math> for all <math> g \in G</math>. | ||
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| + | |||
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'''Links to pages on normal subgroups:''' | '''Links to pages on normal subgroups:''' | ||
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(2) http://eom.springer.de/N/n067690.htm | (2) http://eom.springer.de/N/n067690.htm | ||
| − | (3 | + | (3) http://math.ucr.edu/home/baez/normal.html |
| − | + | ||
| − | + | ||
'''References:''' | '''References:''' | ||
| + | |||
| + | - http://www.math.uiuc.edu/~r-ash/Algebra/Chapter1.pdf | ||
- Dummit, D.S. & Foote, R.M. (1991). Abstract Algebra. United States: Prentice Hall. | - Dummit, D.S. & Foote, R.M. (1991). Abstract Algebra. United States: Prentice Hall. | ||
Revision as of 13:33, 27 April 2011
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
