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+ | [[Category:bonus point project]] | ||
+ | [[Category:math]] | ||
+ | [[Category:MA453]] | ||
+ | [[Category:algebra]] | ||
+ | [[Category:tutorial]] | ||
+ | |||
+ | =Morphisms= | ||
+ | A student project for [[MA453]]: "Abstract Algebra" | ||
+ | ---- | ||
For the following definitions, Let G and H be two groups: | For the following definitions, Let G and H be two groups: | ||
A '''morphism''',<math>\rho\,\!</math>, from G to H is a function <math>\rho\,\!</math>: G --> H such that: | A '''morphism''',<math>\rho\,\!</math>, from G to H is a function <math>\rho\,\!</math>: G --> H such that: | ||
− | 1) | + | 1)<math>I_G</math> = <math>I_H</math> |
2)<math>\rho\,\!</math>(g*g') = <math>\rho\,\!</math>(g)*<math>\rho\,\!</math>(g'), this preserves the multiplication table | 2)<math>\rho\,\!</math>(g*g') = <math>\rho\,\!</math>(g)*<math>\rho\,\!</math>(g'), this preserves the multiplication table | ||
Line 10: | Line 19: | ||
1)Associativity: h o (g o f) = (h o g)o f whenever the operations are defined | 1)Associativity: h o (g o f) = (h o g)o f whenever the operations are defined | ||
2)Identity: for every object X, the identity morphism on X exists such that for every morphism f: A --> B, | 2)Identity: for every object X, the identity morphism on X exists such that for every morphism f: A --> B, | ||
− | + | <math>id_B</math> o f = f = f o <math>id_A</math>. | |
Types of morphisms: | Types of morphisms: | ||
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o Linear map- this is a homomorphism between two vector spaces | o Linear map- this is a homomorphism between two vector spaces | ||
o Algebra homomorphism- this is a homomorphism between two algebras | o Algebra homomorphism- this is a homomorphism between two algebras | ||
+ | |||
•Properties of elements under homomorphisms: | •Properties of elements under homomorphisms: | ||
Let <math>\Phi\,\!</math> be a homomorphism from a group G to a grou H and let g be and element of G. Then: | Let <math>\Phi\,\!</math> be a homomorphism from a group G to a grou H and let g be and element of G. Then: | ||
1) <math>\Phi\,\!</math> carries the identity of G to the identity of H | 1) <math>\Phi\,\!</math> carries the identity of G to the identity of H | ||
− | 2)<math>\Phi\,\!</math>(g^n) = (<math>\Phi\,\!</math>(g))^n for all n in Z | + | 2)<math>\Phi\,\!</math>(<math>g^n</math>) = (<math>\Phi\,\!</math><math>(g))^n</math> for all n in Z |
3)If |g| is finite, then |<math>\Phi\,\!</math>(g)| divides |g| | 3)If |g| is finite, then |<math>\Phi\,\!</math>(g)| divides |g| | ||
4)Ker(<math>\Phi\,\!</math>) is a subgroup of G | 4)Ker(<math>\Phi\,\!</math>) is a subgroup of G | ||
5)aKer(<math>\Phi\,\!</math>) = bKern(<math>\Phi\,\!</math>) if and only if <math>\Phi\,\!</math>(a) = <math>\Phi\,\!</math>(b) | 5)aKer(<math>\Phi\,\!</math>) = bKern(<math>\Phi\,\!</math>) if and only if <math>\Phi\,\!</math>(a) = <math>\Phi\,\!</math>(b) | ||
− | 6)If <math>\Phi\,\!</math>(g) = g' then <math>\Phi\,\!</math> | + | 6)If <math>\Phi\,\!</math>(g) = g' then <math>\Phi\,\!</math>(g') = {x in G | <math>\Phi\,\!</math>(x) = g'} = gKer<math>\Phi\,\!</math> |
•Properties of Subgroups Under Homomorphisms | •Properties of Subgroups Under Homomorphisms | ||
− | Let <math>\Phi\,\!</math> be a homomorphism from a group G to a group | + | Let <math>\Phi\,\!</math> be a homomorphism from a group G to a group <math>\bar{G}</math> and let I be a subgroup of G. Then: |
− | 1)<math>\Phi\,\!</math>(I) = [<math>\Phi\,\!</math>(i) | i in I} is a subgroup of | + | 1)<math>\Phi\,\!</math>(I) = [<math>\Phi\,\!</math>(i) | i in I} is a subgroup of <math>\bar{G}</math> |
2)If I is cyclic, then <math>\Phi\,\!</math>(I) is cyclic | 2)If I is cyclic, then <math>\Phi\,\!</math>(I) is cyclic | ||
3)If I is Abelian, then <math>\Phi\,\!</math>(I) is Abelian | 3)If I is Abelian, then <math>\Phi\,\!</math>(I) is Abelian | ||
4)If I is normal in G, then <math>\Phi\,\!</math>(I) is normal in <math>\Phi\,\!</math>(G) | 4)If I is normal in G, then <math>\Phi\,\!</math>(I) is normal in <math>\Phi\,\!</math>(G) | ||
− | 5)If |Ker<math>\Phi\,\!</math>| = n, then | + | 5)If |Ker<math>\Phi\,\!</math>| = n, then <math>\Phi\,\!</math> is an n-to-1 mapping from G onto <math>\Phi\,\!</math>(G) |
6)If |I| = n, then |<math>\Phi\,\!</math>(I)| divides n | 6)If |I| = n, then |<math>\Phi\,\!</math>(I)| divides n | ||
− | 7)If | + | 7)If <math>\bar{I}</math> is a subgroup of <math>\bar{G}</math>, then <math>\Phi\,\!</math>^-1(<math>\bar{I}</math>) = {i in G | <math>\Phi\,\!</math>(i) in <math>\bar{I}</math>} is a subgroup of G. |
− | 8)If | + | 8)If <math>\bar{I}</math> is a normal subgroup of <math>\bar{G}</math>, then <math>\Phi\,\!</math>^-1(<math>\bar{I}</math>) = {i in G| <math>\Phi\,\!</math>(i) in <math>\bar{I}</math>} is a normal subgroup of G |
− | 9)If <math>\Phi\,\!</math> is onto and | + | 9)If <math>\Phi\,\!</math> is onto and Ker<math>\Phi\,\!</math> = {e}, then <math>\Phi\,\!</math> is an isomorphism from G to <math>\bar{G}</math>. |
− | Examples | + | |
+ | '''Examples''' | ||
• Any isomorphism is a homomorphism that is also onto and 1-to-1 | • Any isomorphism is a homomorphism that is also onto and 1-to-1 | ||
− | • The mapping | + | • The mapping <math>\Phi\,\!</math> from Z to <math>Z_n</math>, definded by <math>\Phi\,\!</math>(m) = m mod n is a homomorphism |
− | • The mapping | + | • The mapping <math>\Phi\,\!</math>(x) = <math>x^2</math> from R*, the nonzero real numbers under multiplication, to itself is a homomorphism. This is because <math>\Phi\,\!</math>(ab) =<math>(ab)^2</math> = <math>a^2b^2</math> = <math>\Phi\,\!</math>(a)<math>\Phi\,\!</math>(b) for all a and b in R* |
− | • The exponential function rho : x --> e^x is an isomorphism. It is injective (monomorphism) and surjective (epimorphism) because one can take logs. | + | • The exponential function <math>\rho\,\!</math> : x --> <math>e^x</math> is an isomorphism. It is injective (monomorphism) and surjective (epimorphism) because one can take logs. |
+ | |||
+ | |||
+ | • : (<math>R_t</math> , *) --> (<math>R_t</math> , *) is an isomorphism | ||
− | |||
• ( *2) : Z/3Z --> Z/3Z is a monomorphism,epimorphism and isomorphism | • ( *2) : Z/3Z --> Z/3Z is a monomorphism,epimorphism and isomorphism | ||
+ | |||
+ | |||
+ | '''Links to University Pages''' | ||
+ | |||
+ | • http://astarmathsandphysics.com/university_maths_notes/abstract_algebra_and_group%20theory/university_maths_notes_abstract_algebra_group_theory_morphisms.html | ||
+ | |||
+ | • http://www.math.purdue.edu/~mdd/Publications/Qd-morphisms-JFA.pdf | ||
+ | |||
+ | • http://www.math.purdue.edu/~lipshitz/cexprintss.pdf | ||
+ | |||
+ | • http://www.math.purdue.edu/~mdd/Publications/shape.pdf | ||
+ | |||
+ | • http://www.math.purdue.edu/~mdd/Publications/A.pdf | ||
+ | |||
+ | • https://www.projectrhea.org/rhea/index.php/NotesWeek4Th_MA453Fall2008walther | ||
+ | |||
+ | '''Other interesting pages on Morphisms''' | ||
+ | |||
+ | • http://www.jstor.org/stable/3481861?seq=2 | ||
+ | |||
+ | • http://www.math.columbia.edu/~scautis/papers/pmm.pdf | ||
+ | |||
+ | '''References''' | ||
+ | |||
+ | • http://en.wikipedia.org/wiki/Morphism | ||
+ | |||
+ | • Gallian, J.A. (2010). Contemporary Abstract Algebra. United States: Brooks/Cole. | ||
+ | |||
+ | • MA 453 class notes, Professor Walther Lecture |
Latest revision as of 10:16, 21 March 2013
Morphisms
A student project for MA453: "Abstract Algebra"
For the following definitions, Let G and H be two groups:
A morphism,$ \rho\,\! $, from G to H is a function $ \rho\,\! $: G --> H such that:
1)$ I_G $ = $ I_H $ 2)$ \rho\,\! $(g*g') = $ \rho\,\! $(g)*$ \rho\,\! $(g'), this preserves the multiplication table
The domain and the codomain are two operations that are defined on every morphism.
Morphims satisfy two axioms: 1)Associativity: h o (g o f) = (h o g)o f whenever the operations are defined 2)Identity: for every object X, the identity morphism on X exists such that for every morphism f: A --> B, $ id_B $ o f = f = f o $ id_A $.
Types of morphisms:
An epimorphism is a morphism where for every h in H, there is at least one g in G with f(g) = h
•This is the same as saying that $ \rho\,\! $ is surjective or onto
A monomorphism is a morphism for which $ \rho\,\! $(g) = $ \rho\,\! $(g') can only happen if g = g'
•This is the same as saying that $ \rho\,\! $ is injective
An isomorphism is a morphism that is both an epimorphism and a monomorphism (both surjective and injective). This means that $ \rho\,\! $ sets up a 1-to-1 correspondence between the elements of G and the elements of H.
•This is the same as saying that $ \rho\,\! $ is bijective
An automorphism is an isomorphism from a function to itself. It is a way of mapping the object to itself while preserving all of its structure.
•An inner automorphism is a function ƒ: G → G such that ƒ(x) = a−1xa, for all x in G, where a is a given fixed element of G.
A homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces).
•Types of homomorphisms: o Group homomorphism- this is a homomorphism between two groups. o Ring homomorphism- this is a homomorphism between two rings. o Functor- this is a homomorphism between two categories o Linear map- this is a homomorphism between two vector spaces o Algebra homomorphism- this is a homomorphism between two algebras
•Properties of elements under homomorphisms: Let $ \Phi\,\! $ be a homomorphism from a group G to a grou H and let g be and element of G. Then: 1) $ \Phi\,\! $ carries the identity of G to the identity of H 2)$ \Phi\,\! $($ g^n $) = ($ \Phi\,\! $$ (g))^n $ for all n in Z 3)If |g| is finite, then |$ \Phi\,\! $(g)| divides |g| 4)Ker($ \Phi\,\! $) is a subgroup of G 5)aKer($ \Phi\,\! $) = bKern($ \Phi\,\! $) if and only if $ \Phi\,\! $(a) = $ \Phi\,\! $(b) 6)If $ \Phi\,\! $(g) = g' then $ \Phi\,\! $(g') = {x in G | $ \Phi\,\! $(x) = g'} = gKer$ \Phi\,\! $ •Properties of Subgroups Under Homomorphisms Let $ \Phi\,\! $ be a homomorphism from a group G to a group $ \bar{G} $ and let I be a subgroup of G. Then: 1)$ \Phi\,\! $(I) = [$ \Phi\,\! $(i) | i in I} is a subgroup of $ \bar{G} $ 2)If I is cyclic, then $ \Phi\,\! $(I) is cyclic 3)If I is Abelian, then $ \Phi\,\! $(I) is Abelian 4)If I is normal in G, then $ \Phi\,\! $(I) is normal in $ \Phi\,\! $(G) 5)If |Ker$ \Phi\,\! $| = n, then $ \Phi\,\! $ is an n-to-1 mapping from G onto $ \Phi\,\! $(G) 6)If |I| = n, then |$ \Phi\,\! $(I)| divides n 7)If $ \bar{I} $ is a subgroup of $ \bar{G} $, then $ \Phi\,\! $^-1($ \bar{I} $) = {i in G | $ \Phi\,\! $(i) in $ \bar{I} $} is a subgroup of G. 8)If $ \bar{I} $ is a normal subgroup of $ \bar{G} $, then $ \Phi\,\! $^-1($ \bar{I} $) = {i in G| $ \Phi\,\! $(i) in $ \bar{I} $} is a normal subgroup of G 9)If $ \Phi\,\! $ is onto and Ker$ \Phi\,\! $ = {e}, then $ \Phi\,\! $ is an isomorphism from G to $ \bar{G} $.
Examples
• Any isomorphism is a homomorphism that is also onto and 1-to-1
• The mapping $ \Phi\,\! $ from Z to $ Z_n $, definded by $ \Phi\,\! $(m) = m mod n is a homomorphism
• The mapping $ \Phi\,\! $(x) = $ x^2 $ from R*, the nonzero real numbers under multiplication, to itself is a homomorphism. This is because $ \Phi\,\! $(ab) =$ (ab)^2 $ = $ a^2b^2 $ = $ \Phi\,\! $(a)$ \Phi\,\! $(b) for all a and b in R*
• The exponential function $ \rho\,\! $ : x --> $ e^x $ is an isomorphism. It is injective (monomorphism) and surjective (epimorphism) because one can take logs.
• : ($ R_t $ , *) --> ($ R_t $ , *) is an isomorphism
• ( *2) : Z/3Z --> Z/3Z is a monomorphism,epimorphism and isomorphism
Links to University Pages
• http://www.math.purdue.edu/~mdd/Publications/Qd-morphisms-JFA.pdf
• http://www.math.purdue.edu/~lipshitz/cexprintss.pdf
• http://www.math.purdue.edu/~mdd/Publications/shape.pdf
• http://www.math.purdue.edu/~mdd/Publications/A.pdf
• https://www.projectrhea.org/rhea/index.php/NotesWeek4Th_MA453Fall2008walther
Other interesting pages on Morphisms
• http://www.jstor.org/stable/3481861?seq=2
• http://www.math.columbia.edu/~scautis/papers/pmm.pdf
References
• http://en.wikipedia.org/wiki/Morphism
• Gallian, J.A. (2010). Contemporary Abstract Algebra. United States: Brooks/Cole.
• MA 453 class notes, Professor Walther Lecture