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Group Homomorphisms
A homomorphism between groups (G, \cdot) and (H, \circ) is a map \phi :G \rightarrow H such that \[\begin{aligned}\end{aligned}\] for g_1, g_2 \in G. The range of \phi in H is called the homomorphic image of \phi.
Group Homomorphisms
A homomorphism between groups \((G, \cdot)\) and \((H, \circ)\) is a map \(\phi :G \rightarrow H\) such that \[\begin{aligned}\end{aligned}\] for \(g_1, g_2 \in G\). The range of \(\phi\) in \(H\) is called the homomorphic image of \(\phi\).
Two groups are related in the strongest possible way if they are isomorphic; however, a weaker relationship may exist between two groups. For example, the symmetric group \(S_n\) and the group \({\mathbb Z}_2\) are related by the fact that \(S_n\) can be divided into even and odd permutations that exhibit a group structure like that \({\mathbb Z}_2\), as shown in the following multiplication table.
\[\begin{aligned}\end{aligned}\]
We use homomorphisms to study relationships such as the one we have just described.
Example
Let \(G\) be a group and \(g \in G\). Define a map \(\phi : {\mathbb Z} \rightarrow G\) by \(\phi( n ) = g^n\). Then \(\phi\) is a group homomorphism, since \[\begin{aligned}\end{aligned}\]. This homomorphism maps \({\mathbb Z}\) onto the cyclic subgroup of \(G\) generated by \(g\).
Example
Let \(G = GL_2( {\mathbb R })\). If \[\begin{aligned}\end{aligned}\] is in \(G\), then the determinant is nonzero; that is, \(\det(A) = ad - bc \neq 0\). Also, for any two elements \(A\) and \(B\) in \(G\), \(\det(AB) = \det(A) \det(B)\). Using the determinant, we can define a homomorphism \(\phi : GL_2( {\mathbb R }) \rightarrow {\mathbb R}^\ast\) by \(A \mapsto \det(A)\).
Example
Recall that the circle group \({ \mathbb T}\) consists of all complex numbers \(z\) such that \(|z|=1\). We can define a homomorphism \(\phi\) from the additive group of real numbers \({\mathbb R}\) to \({\mathbb T}\) by \(\phi : \theta \mapsto \cos \theta + i \sin \theta\). Indeed, \[\begin{aligned}\phi( \alpha + \beta ) & = \cos( \alpha + \beta ) + i \sin( \alpha + \beta ) \\ & = (\cos \alpha \cos \beta - \sin \alpha \sin \beta) + i( \sin \alpha \cos \beta + \cos \alpha \sin \beta ) \\ & = (\cos \alpha + i \sin \alpha )(\cos \beta + i \sin \beta) \\ & = \phi( \alpha ) \phi( \beta )\end{aligned}\]. Geometrically, we are simply wrapping the real line around the circle in a group-theoretic fashion.
The following proposition lists some basic properties of group homomorphisms.
Condensed: the full section is in Judson, Abstract Algebra: Theory and Applications.
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Questions people ask
What is a group, in plain words?
A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group. That is where the idea came from.
What is the difference between a ring and a field?
A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.
Parts of this page are adapted from Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). Condensed and re-explained here; errors are ours.
More in Abstract Algebra
GroupsSubgroups, cosets and Lagrange's theoremCyclic groups and permutation groupsHomomorphisms, normal subgroups and quotient groupsRings and fieldsGalois theory: why the quintic has no formula