Homomorphism, Mathematics

Let G be a group acting on a set X. The action is called faithful if for any g ≠ 1 ∈ G there exists an x ∈ X such that gx ≠ x. That is, only the identity fi xes everything.

Prove the following:

(a) A group G acts faithfully on X if and only if the corresponding homomorphism  Ψ: G -> AutSet(X) is injective. Thus, for a faithful action, G is isomorphic to a subgroup of AutSet(X).

(b) In general, G/ker Ψ acts faithfully on X. (You will need to de ne how G= ker Ψ acts, and make sure this is an action.)

Let p, q and l be 3 distinct prime numbers, and let G be a finite group. Prove that G is solvable if:

991_Homomorphism.png

Posted Date: 2/26/2013 2:55:10 AM | Location : United States







Related Discussions:- Homomorphism, Assignment Help, Ask Question on Homomorphism, Get Answer, Expert's Help, Homomorphism Discussions

Write discussion on Homomorphism
Your posts are moderated
Related Questions

Direction Cosines This application of the dot product needs that we be in three dimensional (3D) space not like all the other applications we have looked at to this point.

Let 0 ! V1 !    ! Vk ! 0 be a long exact sequence of vector spaces with linear maps. Show that P (??1)i dim Vi = 0.

1. Let S be the set of all nonzero real numbers. That is, S = R - {0}. Consider the relation R on S given by xRy iff xy > 0. (a) Prove that R is an equivalence relation on S, an

A ?ight from Pittsburgh to Los Angeles took 5 hours and covered 3,060 miles. What was the plane's average speed? Find out the rate at that Susan is traveling through dividing h

The formula for the surface area of a sphere is 4πr 2 . What is the surface area of a ball with a diameter of 6 inches? Round to the nearest inch. (π = 3.14) If the diameter  o


How can I submit a sample of my work in either teaching online or checking homework as I am retired and doing this for the first time?

Use an appropriate infinite series method about x = 0 to find two solutions of the given differential equation: y''''-xy''-y=0

What is Multiplying Fractions ? The rule for multiplying fractions is to "multiply across": Multiply the numerators to get the numerator of the answer. Multiply the den