Two questions in group theory
Not what you're looking for?
Problem #1
Prove that Aut(V)= (S3)and that Aut(S3)= S3.
Problem #2
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).
Purchase this Solution
Solution Summary
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.
Solution Preview
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 .
Label
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 ...
Purchase this Solution
Free BrainMass Quizzes
Exponential Expressions
In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.
Geometry - Real Life Application Problems
Understanding of how geometry applies to in real-world contexts
Solving quadratic inequalities
This quiz test you on how well you are familiar with solving quadratic inequalities.
Know Your Linear Equations
Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.
Graphs and Functions
This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.