- #1
jstrunk
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- TL;DR Summary
- 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?
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?