Purchase Solution

Groups and proofs

Not what you're looking for?

Ask Custom Question

I)
Let A, B, and C be groups, let alpha, beta, and gamma be homomorphisms with gamma times alpha = beta

alpha gamma beta
A--------->B----------->C<---------A

If alpha is surjective, prove that ker(gamma)= alpha((ker(beta)).

ii)
Prove that if K is a subgroup of a group G, and if every left coset of aK is equal to the right coset Kb, then K is a normal subgroup of G.

I just dont know how to do these, the confuse me and I am horrible at writing them in good proof form

Purchase this Solution

Solution Summary

This contains proofs regarding a surjective homomorphism and a normal subgroup.

Solution Preview

Please see the attachment.

Problem #1
Proof:
From the condition, we have the following relations.
,
We have .
Now suppose is surjective, we want to show that
For any , we have , then . ...

Purchase this Solution


Free BrainMass Quizzes
Probability Quiz

Some questions on probability

Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

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.

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.