hello all, my question is what is the outer automorphism of the alternating group A4, and a brief explanation why. I've heard/read somewhere that it's the factor group S4/A4, but i'm not too sure about that (or know why)

Any help would be much appreciated, thanks.

Not entirely sure what you're asking:

- The automorphism group of the alternating group A4 is the symmetric group S4

- The inner automorphism group of A4 is again A4

- The outer automorphism group is by definition the quotient of the automorphism group by the inner automorphism group, which for A4 is S4/A4, which is the group Z2 because every element of S4 is either an odd or even permutation.

Your question is about one of these aspects?

Personally I would like to understand the geometry of all this. Think of S4 as the symmetry group of a regular tetrahedron. Then, for example, any rotation is an even permutation and so is a member of A4. On the other hand, every reflection is an odd permutation. But not every symmetry is a reflection or rotation and I would like to have a clearer picture of which other symmetries are even. I fould something on Wiki about even permutations being those symmetries that are "orientation preserving" but I didn't understand it.

yeah it was that. Unfortunately I have just had the exam so i don't need to know anymore. Thanks a lot for your help though!

Since this is SMP, this is sort of appropriate. Many of us may have studied group theory during a math course, and a few of us may have studied group theory in a science course (chemists and physicists I think?). However, I once studied group theory in a philosophy course. It was in a course in symbolic logic, and we basically just covered a few nearly trivial results I already knew from my math courses, but I just found the whole idea of studying group theory in philosophy to be interesting.

Don't forget geologists, mickey! Learning all the possible symmetries of crystals, and being able to identify an example of any of the 32 point groups on sight, is a standard part of sophomore mineralogy. I have to say my mineralogy professor explained it better than my abstract algebra teacher two years later did, too...