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#### AutGuy98

##### New member

- Sep 11, 2019

- 20

Hey guys,

I have some more problems that I need help with figuring out what to do. The second (and final) one is divided into 4 mini-problems, sub-sections, whatever you would like to call them. It asks:

(a) Show that the set of automorphisms of a group G, denoted by Aut(G), is a group under the usual composition of functions.

(b) Let G be a group and $g\in G$. Define a map $\psi_g:G\to G$ as follows: for any $h\in G, \psi_g(h)=ghg^{-1}$. Show that $\psi_g$ is an automorphism.

(c) Show that the map $\gamma:G\to Aut(G),\,g\mapsto\psi_g$ is an homomorphism of groups and compute its kernel.

(d) Let $\gamma(G)=H=\{\psi_g \mid g\in G\}$ (one usually refers to H as the group of inner automorphisms of G). Show that H is a normal subgroup of Aut(G).

I would greatly appreciate it if someone could please get back to me about these by tomorrow. But if more time is needed to work the problems out, I completely understand. Thank you in advance to whomever assists me with these. I am extremely grateful.

I have some more problems that I need help with figuring out what to do. The second (and final) one is divided into 4 mini-problems, sub-sections, whatever you would like to call them. It asks:

(a) Show that the set of automorphisms of a group G, denoted by Aut(G), is a group under the usual composition of functions.

(b) Let G be a group and $g\in G$. Define a map $\psi_g:G\to G$ as follows: for any $h\in G, \psi_g(h)=ghg^{-1}$. Show that $\psi_g$ is an automorphism.

(c) Show that the map $\gamma:G\to Aut(G),\,g\mapsto\psi_g$ is an homomorphism of groups and compute its kernel.

(d) Let $\gamma(G)=H=\{\psi_g \mid g\in G\}$ (one usually refers to H as the group of inner automorphisms of G). Show that H is a normal subgroup of Aut(G).

I would greatly appreciate it if someone could please get back to me about these by tomorrow. But if more time is needed to work the problems out, I completely understand. Thank you in advance to whomever assists me with these. I am extremely grateful.

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