# A very challenging question regarding in basic algebra group theory?

• jessicaw
In summary: So in order for every function to be an isomorphism from G to G, G must have only one element, making it a trivial group.
jessicaw
1.Why $Aut(G)=S_G$ implies G is trivial?
I search through the internet and no answer.2.Here is another very difficult conception question which has different answers from my professor and wikipedia:
Difference between Symmetry group,automorphism group and Permutation group?
From wikipedia:automorphism group is, "loosely" speaking, the symmetry group of the object.
My guess is Symmetry group equals Permutation group? and automorphism group is smaller?

Last edited:
The automorphism group is the set of functions which are group isomorphisms from G to G. So if we have the group Z4, the function f(x) with f(0)=0, f(1)=3, f(2)=2 and f(3)=1 is a group automorphism. The function with f(0)=0, f(1)=2, f(2)=0, f(3)=2 is not a group automorphism, because it's not an isomorphism, nor is f(0)=0, f(1)=2, f(2)=1, f(3)=3 because this isn't even a homomorphism.

Loosely speaking, you could say automorphism is a way to swap group elements that act equivalent: we can swap the 1 with the 3 because both are generators of Z4 (f(1)=3), but we can't swap the 1 and the 2 because they have different orders (f(1)=2).

SG is the set of all functions on G, which is sometimes called the permutation group of G because a permutation is just a general way to swap elements of G.

Symmetry groups refer to symmetries of geometric objects. Unless you have a geometric representation for your group it's not a symmetry group (for example, the set of reflections/rotations that take the unit square to itself in the plane forms a symmetry group)

So the question is asking why, if every function is an isomorphism from G to G, is G the trivial group? Think about properties that isomorphisms must have

jessicaw said:
1.Why $Aut(G)=S_G$ implies G is trivial?

In short, because every group has a unique identity element.

## 1. What is group theory in basic algebra?

Group theory is a branch of mathematics that deals with the study of groups, which are sets of elements with a binary operation that satisfies certain properties. In basic algebra, group theory is used to understand the structure and properties of algebraic structures, such as rings and fields.

## 2. What makes a question in basic algebra group theory challenging?

A question in basic algebra group theory can be challenging due to its complexity and the level of abstraction involved. It may require a deep understanding of abstract concepts and the ability to apply them in a variety of situations.

## 3. How can I improve my understanding of basic algebra group theory?

To improve your understanding of basic algebra group theory, it is important to have a solid foundation in basic algebra and abstract algebra. You can also practice solving problems and working through proofs to strengthen your understanding.

## 4. What are some real-world applications of group theory in basic algebra?

Group theory has many real-world applications, especially in the fields of physics, chemistry, and computer science. For example, it is used to study the symmetries of molecules in chemistry and to analyze the structure of crystals in physics.

## 5. Are there any resources available for learning more about basic algebra group theory?

Yes, there are many resources available for learning more about basic algebra group theory. You can find textbooks, online courses, and video lectures that cover the topic. It can also be helpful to join study groups or seek guidance from a mentor or tutor.

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