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Homomorphism of groups

, 16:36, 22 June 2012
Commutes with arbitrary words
''[[Homomorphism]] redirect here. For the more general definition of homomorphism, refer [[homomorphism of universal algebras]]''
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==Definition==
===Textbook definition (with symbols)===
Let $G$ and $H$ be [[group]]s. Then a map $\phivarphi: G$ &rarr; $\to H$ is termed a '''homomorphism''' of groups if $\phivarphi$ satisfies the following condition:
$\phivarphi(ab) = \phivarphi(a) \phivarphi(b)$ for all $a, b$ in $G$
===Universal algebraic definition (with symbols)===
Let $G$ and $H$ be [[group]]s. Then a map $\phivarphi:G$ &rarr; $\to H$ is termed a '''homomorphism''' of groups if $\phivarphi$ satisfies ''all'' the following conditions:
* $\phivarphi(ab) = \phivarphi(a) \phivarphi(b)$ for all $a, b$ in $G$* $\phivarphi(e) = e$* $\phivarphi(a^{-1}) = (\phivarphi(a))^{-1}$
===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 $\alpha: G$ &rarr; $\to H$ and $\beta: H$ &rarr; $\to K$ are homomorphisms, then the composite mapping $\beta.\alpha$ is a homomorphism from $G$ to $K$. 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.
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==Kernel and image==
===Expressibility as composite of surjective and injective homomorphism===
Given any homomorphism $\phivarphi: G$ &rarr; $\to H$ of groups, suppose $K = \phivarphi(G)$. Then, we can view $\phivarphi$ as the composite of a surjective homomorphism (viz a [[quotient map]]) from $G$ to $K$ and an injective homomorphism (viz a subgroup inclusion) from $K$ to $H$. 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 $w$ is a [[word]] in the letters $x_1,x_2,\dots,x_n$ (these are just formal symbols). Suppose $\varphi:G \to H$ is a homomorphism of groups. Then, $\varphi$ commutes with $w$, i.e.: $\varphi(w(g_1,g_2,\dots,g_n)) = w(\varphi(g_1),\varphi(g_2),\dots,\varphi(g_n)) \ \forall \ g_1,g_2,\dots,g_n \in G$ where the $w$ on the left is the [[word map]] in $G$ (i.e., it evaluates the word for a tuple of values of the letters in $G$ and the $w$ on the right is the word map in $H$.
===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.
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