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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.
Manje Akukho kalkuli yokuxazulula le nto, kodwa amaphuzu ayo anga kalkuli. Zama enye ngezansi, noma ubhale yakho.
Amaphawu asetshenziswa lapha
Chofoza noma yiluphi uphawu lwesibonakaliso ukuze uthole ukuchaza okuphelele, isithombe, kanye nokuthi iyiphi inhlamvu ekhona ekhuluma ngaso.
Imibuzo abantu bebuza
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.
Amangxenye wale khasi aguqulwe kusuka Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). I-condensed futhi iphinde icaciswe lapha; amaphutha awusizo.
Okuningi 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