# A question about the search for normal subgroups- Conjugacy classes ?

• nalkapo
In summary, the conversation is about finding the number of conjugacy classes in Aut(Z*_24) and using it to show that Aut(Z*_24) is simple. The definition of a simple group is also discussed, as well as the relationship between conjugacy classes and normal subgroups. The total number of elements in Aut(Z*_24) is mentioned, and the question of how conjugacy classes are computed is raised. It is suggested that there may be a connection between nontrivial normal subgroups and conjugacy classes.
nalkapo
A question about the search for normal subgroups- Conjugacy classes!?

## Homework Statement

Aut(Z*_24) has Conjugacy classes of order 1, 21, 24, 24, 42 and 56.
Show that Aut(Z*_24) is simple.

Note: Aut(Z*_24) is sometimes written as U(Z_24)
Thanks for any idea or answer...

## The Attempt at a Solution

I have no idea! At first, why 24 is written 2 times?

It means that it has six conjugacy classes, two of which have the order 24. For starters, what's the definition of a simple group? And how do conjugacy classes relate to that definition?

Simple group means, if it contains no non-trivial normal subgroups. so, for example, if a group has prime order, then it has no nontrivial normal subgroup.
and conjugacy classes means for a and b in G, there exist an element g for which,
g.a.g^(-1)=b
that means g.a=g.b
also i know that, Aut(Z*_24) has 168 elements and this is the total of this conjugacy classes. for next step, what should I do?
Actually i want to know, how can I find conjugacy classes? how did they find this number of conjugacy classes?

So is there perhaps a connection between the normal subgroups and conjugacy classes? In particular, what would it mean in terms of conjugacy classes, if you had a nontrivial normal subgroup?

I don't understand what is conjugacy classes and how to compute them.. only i know its definiton.

## 1. What is a normal subgroup?

A normal subgroup is a subgroup of a group that is invariant under conjugation, meaning that for any element in the subgroup, its conjugates by all other elements in the group are also in the subgroup.

## 2. How do you find the normal subgroups of a group?

To find the normal subgroups of a group, you can use the conjugacy class equation, which states that the number of conjugacy classes of a group is equal to the index of its center. The center of a group is the set of elements that commute with all other elements in the group. Normal subgroups can also be found by examining the group's structure and identifying any subgroups that are invariant under conjugation.

## 3. Can a normal subgroup be non-abelian?

Yes, a normal subgroup can be non-abelian. The normality of a subgroup is related to its invariance under conjugation, not its commutativity. A subgroup can be non-abelian and still be invariant under conjugation, making it a normal subgroup.

## 4. How are normal subgroups related to quotient groups?

Normal subgroups are closely related to quotient groups. In fact, the quotient group is defined as the group of cosets of a normal subgroup. This means that the elements of the quotient group are the left or right cosets of the normal subgroup, and the group operation is defined by coset multiplication. Normal subgroups can also be used to define the kernel of a homomorphism, which is a fundamental concept in group theory.

## 5. Can every subgroup be a normal subgroup?

No, not every subgroup can be a normal subgroup. A subgroup must satisfy certain conditions to be classified as a normal subgroup, including being invariant under conjugation. If a subgroup does not meet these conditions, it cannot be a normal subgroup.

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