Problems in Group Theory
Not what you're looking for?
See the attached file.
Suppose that G is a finite group and H is a normal subgroup of G.
(i) Explain briefly how the quotient group G/H is defined. [You should explain what the elements of G/H are, how the multiplication is defined, what the identity element is and how inverses are defined. You do not need to prove that the multiplication is well-defined.]
Show that the map
?:G?G/H
?(g)=gH
is a homomorphism. What is the kernel of ??
(ii) Hence show that if ?:G/H?GL(n,C) is a representation of the quotient group G/H then the map ? ?:G?GL(n,C) defined by
? ?(g)=?(gH)
(for each g?G) is a representation of G.
(iii) Now let G=A_4, the alternating group of even permutations of {1,2,3,4}. Let
H={e,(12)(34),(13)(24),(14)(23)}
Show that H?G. To which well-known group is the quotient group G/H isomorphic?
(iv) For G and H as in (iii) above, write down (without proof) three degree one representations of G/H.
(v) Hence describe three degree 1 (and hence irreducible) representations of G=A_4. Write down the characters of the representations you have found on representatives of the four conjugacy classes of G.
Purchase this Solution
Solution Summary
The solution solves several problems in group theory involving such topics as group representations, alternating groups, permutation groups, group homomorphisms, conjugacy classes, normal subgroups, general linear groups, and characters.
Purchase this Solution
Free BrainMass Quizzes
Probability Quiz
Some questions on probability
Graphs and Functions
This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.
Exponential Expressions
In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.
Multiplying Complex Numbers
This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.
Solving quadratic inequalities
This quiz test you on how well you are familiar with solving quadratic inequalities.