# 4 problems regarding automorphisms/homomorphisms

• MHB
• AutGuy98
In summary: I've been trying to learn about algebra and calculus and I just don't have the slightest clue where to even start. I was hoping that maybe someone here could be of some help to me in understanding these things and how to do the homework. I also want to apologize for coming off so harshly in my previous post, I was just really frustrated at that point. Again, I do understand where you're coming from and I hope that you can help me out. In summary,I hope someone can help me with these homework problems by tomorrow.
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.

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"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.

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.

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.

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.

Hi Klaas van Aarsen,

## 1. What is an automorphism?

An automorphism is a mathematical function that maps a mathematical structure to itself, while preserving the structure and operations of the original object. In other words, it is a transformation that does not change the essential properties of the object.

## 2. What is the difference between an automorphism and a homomorphism?

While both automorphisms and homomorphisms are mathematical functions, the main difference lies in the type of objects they map. An automorphism maps a mathematical structure to itself, while a homomorphism maps between two different structures, preserving the operations and structure of the original objects.

## 3. What are some applications of automorphisms and homomorphisms?

Automorphisms and homomorphisms have many applications in mathematics and other fields such as physics and computer science. For example, in abstract algebra, they are used to study groups, rings, and fields. In physics, they are used to study symmetries and conservation laws. In computer science, they are used in coding theory and cryptography.

## 4. What are the properties of automorphisms and homomorphisms?

Some key properties of automorphisms and homomorphisms include:

• Preservation of structure and operations
• Injectivity (one-to-one mapping)
• Surjectivity (onto mapping)
• Composition (combining multiple mappings)
• Inverses (a mapping that undoes the original mapping)

These properties help to define and distinguish between different types of automorphisms and homomorphisms.

## 5. Can you give an example of an automorphism and a homomorphism?

An example of an automorphism is the identity function, which maps every element of a set to itself, preserving the structure and operations of the set. An example of a homomorphism is the absolute value function, which maps real numbers to their absolute values, preserving the addition and multiplication operations.

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