Prove that Aut(V)= (S3)and that Aut(S3)= S3.
If H and K are normal subgroups of a group G with HK = G. Prove that
G/(H n K) = (G/H) x (G/K).
1) Let G be a finite group with n elements. We can list the elements of G as . Assume that . Then to any automorphism , we can associate a permutation by the rule where . (We can ignore what happens to the identity since the identity is fixed for all automorphisms, )
In other words, the group of automorphisms is isomorphic to some subgroup of .
a) has 4 elements, so is isomorphic to some subgroup of .
defines an automorphism with associated permutation . Since has order 2, Lagrange's theorem tells us
defines an automorphism with associated permutation . Since has order 3, Lagrange's theorem tells us .
These two facts combined tell us . But , so we must have .
b) Since is generated by ...
The solution comprises approximately 2 pages written in Word with mathematical notation written using Mathtype. Each step is explained, but assumes some background knowledge in abstract algebra. The solution is typical of the sort of tricks that must be employed to answer questions in this field.