MAT102 Lecture 23B

• Homomorphism
  • Functions between groups
  • If $(G,{\oplus }_G)$ and $(H,{\oplus }_H)$ are groups
  • Then $\varphi :G\to H$ is a group homomorphism
  • if $\varphi (g_1{\oplus }_G{g}_2)=\varphi (g_1){\oplus }_H\varphi (g_2)$ for all $g_1,g_2\in G$
  • Add the elements of g, then use the function.
  • Second is use the function on the elements, then add them together.
  • If we have ${\mathbb{Z}}_6\to {\mathbb{Z}}_{12}$, then there are ${12}^6$ possible functions but $6$ homomorphisms
  • Kernel, things that map to the identity element
• Isomorphism
  • This is a homomorphism that's bijective
  • If they're bijective, then $G$ and $H$ are the same group
  • #tk study homomorphisms
• Counterexample to $\left|G\right|=\left|H\right|⟹G\cong H$
  • $G={\mathbb{Z}}_2\times {\mathbb{Z}}_2$ then $\left|G\right|=4$
  • $H={\mathbb{Z}}_4$ then $\left|H\right|=4$
  • But they are not isomorphic
  • If we have an isomorphism of groups, it preserves orders.
  • If $\varphi :G\to H$ is an isomorphism
  • Then $|\varphi (g)|=\left|g\right|$
  • See that orders for $G$ were $1,2,2,2$
  • ${\mathbb{Z}}_4$ has order $2,4,\dots$
  • Because we have a different order, they can't be isomorphic
  • Being abelian is also preserved
• $S_3\ncong {\mathbb{Z}}_6$
  • Order $3\ne 6$
  • ${\mathbb{Z}}_6$ has order $6$
  • but it's not abelian group
• Suppose $G\cong H$ if $K$ is a proper, non-trivial subgroup of $G$, then $\varphi (K)$ is a proper, non-trivial subgroup of $H$
  • If $K$ is a subgroup, $\varphi (K)$ is also a subgroup
  • If the output was trivial, so it all maps to identity.
  • Then that breaks injectivity.
  • $\left|K\right|=|\varphi (K)|$
  • There's a bijection between the subgroups of each $G$ and $H$
• #tk Prove that $S_3\cong {\mathbb{Z}}_2\times {\mathbb{Z}}_3$
  • Show there's an element of order 6
  • Map the generator of each set to each other
• Generate a homomorphism from an abelian group to a non abelian group
  • Take $G={\mathbb{Z}}_2$ and $H=S_5$
  • $\varphi :{\mathbb{Z}}_2\to S_5$
  • $\varphi (0)=()$
  • $\varphi (1)=(12)$
  • $\varphi (1+1)=\varphi (0)=()$
  • $\varphi (1)\varphi (1)=(12)(12)=()$
  • $⟨(12)⟩=\{(),(12)\}\cong {\mathbb{Z}}_2$
  • The subgroups of non-abelian groups can be abelian
  • If $G$ is abelian and $H⊴G$
    • Let $a,b\in H$ then $ab\in G$ so $ab=ba$ in $G$
    • So $ab=ba$ in $H$
• Combinations of homomorphisms is a homomorphism
  • $\gamma (\varphi (a\oplus b))=\gamma (\varphi (a)\oplus \varphi (b))$
  • $=\gamma (\varphi (a))\oplus \gamma (\varphi (b))$
  • Combinations of isomorphisms is also an isomorphism.
• Let $\varphi g(x)=gx{g}^{-1}$
  • Want to show that $\varphi g(xy)=\varphi g(x)\varphi g(y)$
  • LHS: $\varphi g(xy)=gxy{g}^{-1}$
  • RHS: $\varphi g(x)\varphi g(y)=(gx{g}^{-1})(gy{g}^{-1})$
  • $\varphi g(x)\varphi g(y)=gx{e}_{g}y{g}^{-1}$
  • $\varphi g(x)\varphi g(y)=gxy{g}^{-1}$
  • So it's a homomorphism
  • To prove it's an isomorphism, we have to show it has an inverse.
    • For invertibility, see that ${\varphi }_g^{-1}(x)=g^{-1}xg$
    • ${\varphi }_g({\varphi }_g^{-1}(x))={\varphi }_g(g^{-1}xg)$
    • ${\varphi }_g({\varphi }_g^{-1}(x))=g(g^{-1}xg)g^{-1}$
    • ${\varphi }_g({\varphi }_g^{-1}(x))=x$
    • Try the opposite #tk
  • Isomorphism from a group to itself is an Automorphism
• $\mathrm{Aut}(G)=\{\varphi :G\to G\text{Isomorphic}\}$
• $(\mathrm{Aut}(G),\circ )$ is a group
• $\mathrm{Inn}(G)\{\varphi g:g\in G\}$
  • Inn is inner automorphisms
• $G\cong \mathrm{Inn}(G)⊴\mathrm{Aut}(G)$
• Calculus and groups can combine and is a field called lie groups
• #tk study groups more for exams