Question: Let G and H be two groups, f : G H a group morphism and K G a distinguished subgroup. Show that the following two

Let G and H be two groups, f : G H a group morphism and K G a distinguished subgroup. Show that the following two conditions are equivalent: (A) K Ker f (i.e. For all k K, f(k) = eH, the neutral of H). (B) There exists a group morphism : G/K H such that (x.K) = f(x).

Step by Step Solution

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock blur-text-image

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Students Have Also Explored These Related Mathematics Questions!

Please Solve This Question 1. If f : G - G' is a homomorphism then show that f (G) = (f (a) | a E G} is a subgroup of G'. We also write f(G) = Imf (Image f). Show further that if H is normal in G...

Journal of Autism and Developmental Disorders, Vol. 32, No. 3, June 2002 ( 2002) Descriptive Epidemiology of Autism in a California Population: Who Is at Risk? Lisa A. Croen,1,3 Judith K. Grether,1...

Please solve this question 47. Suppose that g and h induce the same inner automorphism of a group G. Prove that h-'g E Z(G). 48. Combine the results of Exercises 45 and 47 into a single "if and only...

Modern Algebra 1, MATH 5410 Homework 6, Section I.8, Solutions Due Friday, October 23 at 1:40 Write in complete sentences!!! Explain what you are doing and convince me that you understand what you...

Suppose that g and h induce the same inner automorphism of a group G. Prove that h-lg E Z(G). 48. Combine the results of Exercises 45 and 47 into a single "if and only if theorem. 49. If a and B are...

I need these answers for a study guide please! Please use this form to record your answers and include any work that you would like to show the grader (you can paste in photos or scans of your...

\fCPSC 121: Models of Computation Assignment #4, due Wednesday, November 18th , 2015 at 17:00 Each of the following questions asks you to prove a theorem. When the theorem is stated in English, it...

HW 12, MAT 312/AMS 351: SPRING 2017 Problem 1. (1) Set H = {4a + 6b | a, b Z} Z. Show that H is a subgroup of Z. (2) Is H cyclic? Problem 2. Write down all the cyclic subgroups of the following...

1. A player throws a fair die and simultaneously flips a fair coin. If the coin lands heads, then she wins twice, and if tails, then one-half of the value that appears on the die. Determine her...

QUIZ... Let D be a poset and let f : D D be a monotone function. (i) Give the definition of the least pre-fixed point, fix (f), of f. Show that fix (f) is a fixed point of f. [5 marks] (ii) Show that...

Find all curves with the property that if the normal line is drawn at any point P on the curve, then the part of the normal line between and P the x-axis is bisected by the y-axis.

During the time that heat is not measured by temperature, what do we call it a. Thermal energy b. Sensible heat c. Latent heat d. Ice

An annuity investor is approaching the age at which she may consider annuitizing her contract. She's considering different settlement options and wants her registered representative (RR) to explain...

Some characteristics of the determinants of nominal interest rates are listed as follows. Identify the components (determinants) and the symbols associated with each characteristic: Grade it Now Save...