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Homomorphism of groups

765 bytes added, 16:36, 22 June 2012
Commutes with arbitrary words
''[[Homomorphism]] redirect here. For the more general definition of homomorphism, refer [[homomorphism of universal algebras]]''
<section begin=beginner/>
==Definition==
===Textbook definition (with symbols)===
Let <math>G</math> and <math>H</math> be [[group]]s. Then a map <math>\phivarphi: G</math> &rarr; <math>\to H</math> is termed a '''homomorphism''' of groups if <math>\phivarphi</math> satisfies the following condition:
<math>\phivarphi(ab) = \phivarphi(a) \phivarphi(b)</math> for all <math>a, b</math> in <math>G</math>
===Universal algebraic definition (with symbols)===
Let <math>G</math> and <math>H</math> be [[group]]s. Then a map <math>\phivarphi:G</math> &rarr; <math>\to H</math> is termed a '''homomorphism''' of groups if <math>\phivarphi</math> satisfies ''all'' the following conditions:
* <math>\phivarphi(ab) = \phivarphi(a) \phivarphi(b)</math> for all <math>a, b</math> in <math>G</math>* <math>\phivarphi(e) = e</math>* <math>\phivarphi(a^{-1}) = (\phivarphi(a))^{-1}</math>
===Equivalence of definitions===
{{proofat|[[Equivalence of definitions of group homomorphismof groups]]}}
The textbook definition and universal algebraic definition of homomorphism of groups are equivalent. In other words, for a map between groups to be a homomorphism of groups, it suffices to check that it preserves the binary operation.
==Facts==
===Composition===
If <math>\alpha: G</math> &rarr; <math>\to H</math> and <math>\beta: H</math> &rarr; <math>\to K</math> are homomorphisms, then the composite mapping <math>\beta.\alpha</math> is a homomorphism from <math>G</math> to <math>K</math>. This follows directly from either definition.
===Inverse===
Automorphisms of groups can be viewed as symmetries of the group structure. The collection of automorphisms of a group forms a group under composition and this is termed the [[automorphism group]] of the given group.
<section end=beginner/><section begin=revisit/>
==Kernel and image==
===Expressibility as composite of surjective and injective homomorphism===
Given any homomorphism <math>\phivarphi: G</math> &rarr; <math>\to H</math> of groups, suppose <math>K = \phivarphi(G)</math>. Then, we can view <math>\phivarphi</math> as the composite of a surjective homomorphism (viz a [[quotient map]]) from <math>G</math> to <math>K</math> and an injective homomorphism (viz a subgroup inclusion) from <math>K</math> to <math>H</math>. Moreover, this expression (as composite of a surjective and injective homomorphism) is essentially unique. ===Commutes with arbitrary words=== {{further|[[homomorphism commutes with word maps]]}} Suppose <math>w</math> is a [[word]] in the letters <math>x_1,x_2,\dots,x_n</math> (these are just formal symbols). Suppose <math>\varphi:G \to H</math> is a homomorphism of groups. Then, <math>\varphi</math> commutes with <math>w</math>, i.e.: <math>\varphi(w(g_1,g_2,\dots,g_n)) = w(\varphi(g_1),\varphi(g_2),\dots,\varphi(g_n)) \ \forall \ g_1,g_2,\dots,g_n \in G</math> where the <math>w</math> on the left is the [[word map]] in <math>G</math> (i.e., it evaluates the word for a tuple of values of the letters in <math>G</math> and the <math>w</math> on the right is the word map in <math>H</math>.
===Specification on a generating set===
* If a homomorphism of a group is known on two subgroups of the group, then it is also known on the subgroup generated by them.
<section end=revisit/>
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