Menu Close

G-is-a-group-prove-that-G-Z-G-Inn-G-Where-Inn-G-f-f-G-f-is-an-Automorphism-G-




Question Number 204019 by mnjuly1970 last updated on 04/Feb/24
       G is a group :       prove that :  (G/(Z (G ))) ≅ Inn(G )      Where , Inn(G)= {f ∣ f: G →^(f is an Automorphism)  G}
$$ \\ $$$$\:\:\:\:\:{G}\:{is}\:{a}\:{group}\:: \\ $$$$\:\:\:\:\:{prove}\:{that}\::\:\:\frac{{G}}{{Z}\:\left({G}\:\right)}\:\cong\:{Inn}\left({G}\:\right) \\ $$$$\:\:\:\:{Where}\:,\:{Inn}\left({G}\right)=\:\left\{{f}\:\mid\:{f}:\:{G}\:\overset{{f}\:{is}\:{an}\:{Automorphism}} {\rightarrow}\:{G}\right\} \\ $$$$ \\ $$
Commented by mokys last updated on 04/Feb/24
Commented by mnjuly1970 last updated on 05/Feb/24
thanks alot sir..so nice proof
$${thanks}\:{alot}\:{sir}..{so}\:{nice}\:{proof} \\ $$$$\:\cancel{ } \\ $$
Answered by witcher3 last updated on 04/Feb/24
G→Inn(G)  x→^f_x  gxg^−   x→g→gxg^−   kerf_x ={x∈G∣gxg^− =x}⇔{x∈G∣gx=xg}=Z(G)  use isomoprohism Theorem⇒ f surjective  G→^f G′⇒(G/(kerf(G)))≊G′  (G/(ker(f)))≃Im(f)⇒(G/(Z(G)))≊Inn(G)
$$\mathrm{G}\rightarrow\mathrm{Inn}\left(\mathrm{G}\right) \\ $$$$\mathrm{x}\overset{\mathrm{f}_{\mathrm{x}} } {\rightarrow}\mathrm{gxg}^{−} \\ $$$$\mathrm{x}\rightarrow\mathrm{g}\rightarrow\mathrm{gxg}^{−} \\ $$$$\mathrm{kerf}_{\mathrm{x}} =\left\{\mathrm{x}\in\mathrm{G}\mid\mathrm{gxg}^{−} =\mathrm{x}\right\}\Leftrightarrow\left\{\mathrm{x}\in\mathrm{G}\mid\mathrm{gx}=\mathrm{xg}\right\}=\mathrm{Z}\left(\mathrm{G}\right) \\ $$$$\mathrm{use}\:\mathrm{isomoprohism}\:\mathrm{Theorem}\Rightarrow\:\mathrm{f}\:\mathrm{surjective} \\ $$$$\mathrm{G}\overset{\mathrm{f}} {\rightarrow}\mathrm{G}'\Rightarrow\frac{\mathrm{G}}{\mathrm{kerf}\left(\mathrm{G}\right)}\approxeq\mathrm{G}' \\ $$$$\frac{\mathrm{G}}{\mathrm{ker}\left(\mathrm{f}\right)}\simeq\mathrm{Im}\left(\mathrm{f}\right)\Rightarrow\frac{\mathrm{G}}{\mathrm{Z}\left(\mathrm{G}\right)}\approxeq\mathrm{Inn}\left(\mathrm{G}\right) \\ $$$$ \\ $$$$ \\ $$$$ \\ $$
Commented by mnjuly1970 last updated on 04/Feb/24
 so nice solution sir wicher  thx alot
$$\:{so}\:{nice}\:{solution}\:{sir}\:{wicher} \\ $$$${thx}\:{alot} \\ $$$$\:\: \\ $$
Commented by witcher3 last updated on 04/Feb/24
withe pleasur sir   have a nice day
$$\mathrm{withe}\:\mathrm{pleasur}\:\mathrm{sir}\: \\ $$$$\mathrm{have}\:\mathrm{a}\:\mathrm{nice}\:\mathrm{day} \\ $$$$ \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *