---- Let $\phi:G \to G'$ be a [[group homomorphism]], let $g\in G$, and let $H \leq G$. > [!theorem] Theorem. ([[group homomorphisms preserve structure|Properties of Elements under homomorphism]]) > > | *Property* | Description | |----------------------|---------------------------------------------------------------------------------------------------| | *Identities* | $\phi(e_{G})=e_{G'}$ | | *Powers* | $\phi(g^{n})=\phi(g)^{n}$ | | *Orders (element)* | If $\vert g \vert$ is finite, then $\vert \phi (g) \vert$ [[divides\|divides]] $\vert g \vert$ | | *Orders (group)* | If $\vert G \vert$ is finite, then $\vert \phi(g) \vert$ [[divides\|divides]] $\vert g \vert$ and $\vert G \vert$ | | *Cosets and Kernels* | $\phi (a)=\phi(b) \iff a \ker \phi = b \ker \phi$ | | *Preimages* | If $\phi(g)=g'$, then $\phi ^{-1}(g')=g \ker \phi$ | ^51ed4a > [!theorem] Theorem. ([[group homomorphisms preserve structure|Properties of Subgroups under homomorphism]]) > | *Property* | Description | |----------------------------|-------------------------------------------------------------------------------------------------------| | *Subgroups* | $\phi(H)$ is a [[subgroup]] of $G$ | | *Cyclic* | $H$ is [[cyclic group\|cyclic]] $\implies$ $\phi(H)$ is [[cyclic group\|cyclic]] | | *Abelian* | $H$ is [[abelian group\|abelian]] $\implies$ $\phi(H)$ is [[abelian group\|abelian]] | | *Normal* | $H$ is [[normal subgroup\|normal]] in $G$ $\implies$ $\phi(H)$ is [[normal subgroup\|normal]] in $\phi(G)$| | *Kernel* | If $\vert \ker \phi \vert =n$, then $\phi$ is an [[k-to-1 correspondence\|n-to-1 mapping]] from $G$ onto $\phi(G)$| | *Order* | If $H$ is finite, then $\vert \phi(H) \vert$ [[divides]] $\vert H \vert$ | | *Centers* | $\phi\big(Z(G)\big) \leq Z\big(\phi(G)\big)$ | | *Subgroup Preimages* | If $K' \leq G'$ then $\phi ^{-1}(K') \leq G$ | | *Normal Preimages* | If $K' \trianglelefteq G'$ then $\phi ^{-1}(K) \trianglelefteq G$ | | *isomorphism condition* | If $\phi$ is [[surjection\|surjective]] and $\ker \phi = e$, then $\phi$ is an [[group isomorphism\|isomorphism]] from $G$ to $G'$ | ^d35f2b > [!proof]- Proof. ([[group homomorphisms preserve structure]]) # Properties of Elements **Identities.** See [[group homomorphisms take identities to identities and inverses to inverses]]. **Powers.** For $g \in G$, $\phi(g^{k})=\phi(\overbrace{g g \cdots g}^{k \text{ times }})=\phi(g)\phi(g)\cdots \phi(g) = \phi(g)^{k}. $ **Orders (element).** Suppose $|g|$ is finite, say, $|g|=m$. Consider $|\phi(g)|$. Then using the two properties above, $e_{G'}=\phi(g^{m})=\phi(g)^{m}$. Thus $m$ is a multiple of $|\phi(g)|$, i.e., $|g|=k|\phi(g)|$ for some $k$. **Orders (group).** If $G$ is finite, then $|\phi(g)|$ [[divides]] (by the above) and $|g|$ [[divides]] $G$ ([[Lagrange's Theorem]]). So $|\phi(g)|$ [[divides]] $G$. **Cosets and Kernels.** Suppose $\phi(a)=\phi(b)$. Then $\phi(b)^{-1}\phi(a)=\phi(b^{-1}a)=e$, so $b^{-1}a \in \ker \phi$. We have seen this implies $a \ker \phi=b \ker \phi$. Conversely suppose $a \ker \phi = b \ker \phi$. Then for every $k_{1} \in \ker \phi$, there exists $k_{2} \in \ker \phi$ such that $a k_{1}=bk_{2}$. Now $\phi(a)=\phi(ak_{1})=\phi(bk_{2})=\phi(b)$. # Properties of Subgroups **Subgroups.** Obvious. **Cyclic.** Suppose $H=\langle a \rangle$ is [[cyclic group|cyclic]], such that $h=a^{i}$ for some $i$, for any $h \in H$. Then $\phi(h)=\phi(a^{i})=\phi(a)^{i}$; thus any element in $\phi(H)$ has the form $\phi(a)^{i}$ for some $i$. We already know $\phi(H)$ is a [[subgroup]]— this shows it must be [[cyclic group|cyclic]] with [[generating set of a group|generator]] $\phi(a)$. **Abelian.** Suppose $ab=ba$ for all $a,b \in H$. Then $\phi(a)\phi(b)=\phi(b)\phi(a)$, so $\phi(H)$ is too. **Normal.** Suppose $H$ is [[normal subgroup|normal in]] $G$; i.e., $gHg^{-1}=H$ for all $g \in G$. Since $\phi(gHg^{-1})=\phi(g)\phi(H)\phi(g)^{-1}$, we have $\phi(H)$ is invariant under [[conjugate|conjugation]] by elements of $\phi(G)$. Note that we have NOT said it is [[normal subgroup|normal in]] $G'$! Only in $G$. **Kernel.** Suppose $\ker \phi = n$. The [[coset|cosets]] of $\ker \phi$ [[partition]] $G$, each of size $n$. Now apply the **preimages** property above. ---- #### ----- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as Tag > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```