4 problems regarding automorphisms/homomorphisms

[AutGuy98]

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.

 

[HallsofIvy]

"Show that A is B" is typically a matter of showing that "A" satisfies the definition of "B". For example, given a set and a binary operation on a set (in (a) the set is the set of automorphisms on G and the operation is composition) we must show
1) the operation is associative: a*(b*c)= (a*b)*c.
Suppose a, b, and c are automorphisms on G. Then, for any x in G, a*(b*c)(x)= a(b*c(x))= a(b(c(x))) while (a*b)*c (x)= (a*b)(c(x))= a(b(c(x))) also.

2) there is a identity.
Show that the identity automorphism, f(x)= x, is the identity for the group of automorphisms.

3) Every member of the group has an inverse.
Show that if f(x)= y then g(y)= x is also an automorphism and is the inverse of f.

 

[AutGuy98]

Hi HallsofIvy,

Thank you for responding to my post in such a timely manner. Yes, I do believe the information you have stated is correct. However, seeing as I have no clue where to go or what to do (much less, how to do anything) with these problems, would you and whoever else decides to help me out with them please be so kind and courteous as to provide me with exact steps and calculations that need to be made along the way leading up to, and including, the answer? Again, thank you HallsofIvy for responding, I really do appreciate it. But it is detrimental to me that detailed steps be given. Also, the same deal stands with my other posting of the second part for these questions and with regards to time, I would really appreciate it if anyone could help me with solving them sometime today? Please, this is very important to me and it would help me out a great deal. Thank you in advance and I do not mean to come off as sounding so harsh, I am just a typical guy trying to understand how to do this and by having the steps leading to and including the answer, that is just how I have found that I learn and understand best. So, again, the assistance would be greatly appreciated and I thank you ahead of time.

 

[AutGuy98]

Hey guys,

I just wanted to say that I hope my two posts regarding these exercise set questions will not be forgotten and go unanswered. I pray that this is not the case here and if it is not, I also wanted to say that it would really mean the world to me if someone out there could please respond to both of them by the end of today, since I really need to know how to do them by tomorrow. If someone could please give me some form of an update of any kind to reassure me (since I'm currently freaking out about potentially not getting a response in time), I would appreciate it much more than anybody could know at this point in time. Once again, I say thank you to the person(s) that do reply/respond back and to all those who have helped me with problems in the past. I look forward to reading anything anyone hopefully sends me.

 

[I like Serena]

Hello AutGuy98,

Now that you already have a number of posts here, I'd like to point out that the people here are volunteers.
That is, we are not here to do the homework for other people.
Instead we want to help people out who show some effort but are stuck one way or another.

 

[AutGuy98]

Hi Klaas van Aarsen,

I completely understand where you're coming from with what you said in your previous post and I can see why you would say that. I'll start by saying that while I have posted a lot with little to no work being shown on my own end, the truth is that I just don't understand how to even begin any of the problems. Over these last few years, I found that the only way I can learn how is by examining the steps for each and every one of them, examining why they all work together right up to the answer. It's the only tried and true method that works for me and otherwise, I'm bound to fail every assignment or exercise set that I'm given to do. I really hope that you can see things from my perspective and understand, since the last two posts I made involving four different subsections of questions is something that I have to be able to perform tomorrow in front of my professor and I have no idea how to even start doing them. That is why I come to you guys, because I finally found people that can help me understand the "how", "what", and "why" questions I've had about these problems. Please think about it from this point of view. If you still feel the same way about me and are kind enough to help me complete these exercises first (which would save me completely, otherwise I'm pretty much screwed here), then these will be my final threads here and I will not post anymore. Please, though, I'm begging you guys, anybody here, to help me out here. Thank you in advance, regardless of how things work out.