Undergrad Finding All Automorphisms of Group

[jstrunk]

TL;DR

Finding All Automorphisms of Group

I am very confused about how to find all the automorphisms of a group.
The book I am using is very vague and the exercises don't show any solutions.
I get how to do it for cyclic groups but not the general case.

I will outline what I know of the procedure and insert my questions into it.
To find all automorphisms X:G to G:
a) Pick a generating set S of G
b) Map the identity to the identity
c) Map S to some generating set of G (ie., S itself or one of the other generating sets).
Each element of S must be mapped to an element of the same order.
Question 1: Or do you map each element of S to some element of the same order, without regard to whether it is an element of a
generating set?
d) Apply the homomorphism requirement X(gh)=X(g)X(h) to find the other values X(g) for all g element of G.
Question 2: Do you necessarily get a distinct automorphism for every such distinct mapping?
e) Repeat steps b, c, and d for each generating set (or set of elements of the same order, depending of the answer to Q1).
Question 3: When applying the homomorphism requirement, is it sufficient to find a value X(g) for all g element of G, or do you have
to check that X(gh)=X(g)X(h) for every combination of g,h elements of G to make sure that it is really a valid homomorphism?

Question 4: Is there somewhere online that this subject is explained clearly in an introductory manner?

 

[mathwonk]

Computing all automorphisms of a group seems to be a difficult problem in general, and so I oresume thyere is no procedure for doing so. One class of group automorphisms for non abelian groups are the "inner automorphisms" obtained by conjugating by elements of the group. These give a normal subgroup of the group of automorphisms which is isomorphic to the group itself, modded out by its center. Even for finite p-groups, according to wikipedia, it is unknown whether there is always an automorphism of order p.

https://en.wikipedia.org/wiki/Inner_automorphism

 

[fresh_42]

There should be a way for Lie groups: Calculate the Lie algebra, determine all derivations, which means solving a linear equation system, and see what the Lie group of the derivations is.

Inner automorphisms should be not too difficult, depending how the group is defined. Outer automorphisms sounds difficult to impossible in a reasonable time in general.