- #1

Lee33

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## Homework Statement

Let ##G## be a group of order ##4, G = {e, a, b, ab}, a^2 = b^2 = e, ab = ba.## Determine Aut(G).

**2. The attempt at a solution**

How can I do these types of problems?

When doing these types of problems is any automorphism of ##G## always determined by the images? And why would ##\text{Aut(G)}## be isomorphic to ##S_3?##

I also have a different question. Is the mapping ##T: x\to x^2##, for positive reals under multiplication, an automorphism of its respective group?

I know that it is a bijective but I don't know if its a homomorphism? For homomorphism, I got ##\phi(xy) = (xy)^2## and ##\phi(x)\phi(y) = x^2y^2## but does ##(xy)^2 = x^2y^2##? I know it will be true if it were an abelian group but it doesn't state it.

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