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| + | <nowiki>**Reference citations are denoted throughout as (reference #) after the cited information**</nowiki> | ||
| + | - Mark Knight | ||
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'''Preliminary Definitions''' | '''Preliminary Definitions''' | ||
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The element <math>g</math> ''normalizes'' <math>N</math> if <math>gNg^{-1} = N</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. | + | 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. (reference #2) |
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2. <math>gNg^{-1} = 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. | + | 3. <math>gN = Ng</math> for all <math> g\in G</math>. That is, the left and right cosets are equal. (reference #1) |
| − | 4. <math>N</math> is the kernel of some homomorphism on <math> G </math>. | + | 4. <math>N</math> is the kernel of some homomorphism on <math>G</math>. (reference #2) |
| − | The equivalence of (1), (2) and (3) above is proved | + | The equivalence of (1), (2) and (3) above is proved here: |
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>. | 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>. | ||
| + | For <math> \Leftarrow </math> we have then <math>(aN)(bN) = a(Nb)N = abNN = abN </math>. | ||
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| + | For <math> \Rightarrow </math> then <math>gNg^{-1} \subseteq gNg^{-1}N </math> since <math> 1\in N</math> and by the hypothesis <math>(gN)(g^{-1}N) = gg^{-1}N (=N)</math>. Then we have <math> gNg^{-1} \subseteq N </math> which implies that <math> N\subseteq g^{-1}Ng </math>. Because this result holds for all <math> g \in G</math>, we have <math> N \subseteq gNg^{-1} </math> and the desired result follows. <math> \Box </math> (reference #1) | ||
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| + | ''' Examples of Normal Subgroups ''' | ||
| + | ---- | ||
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| + | 1. Every subgroup of an Abelian group is normal because for elements a in G and h in N, ah = ha. (reference #3) | ||
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| + | 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) | ||
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| + | 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<math>ABA^{-1}</math> = <math>detAdetBdetA^{-1}</math> = detB = 1. (reference #1) | ||
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'''Links to pages on normal subgroups:''' | '''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:''' | '''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 | |
Revision as of 15:13, 27 April 2011
**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)
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
