A very challenging question regarding in basic algebra group theory?

[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?

 

[Office_Shredder]

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).

S_G 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

 

[bpet]

1.Why Aut(G)=S_G implies G is trivial?

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