For groups g_1 and g_2 a function f: g_1 -rightarrow g_2 is a homomorphism if f (g*1 h) = f(g) *2/(h) for every g, h e g1. let k = {g e g1 {(g) = e_2} where e_2 is the identity of g_2. show that k is a subgroup of g1. show that if f is one-to-one then k = {e1} and conversely that if k = {e1} then f must be one-to-one.

You are in charge of the customer service phone line with many employees. at the end of each phone callthe customers are asked to rate their service from 1 to 5 stars, with 5 being the best. you find the averagerating for 20 randomly selected employees and decide you want to improve the ratings so you require thoseemployees to go through a new training program. you then wait a week and find the new average rating foreach employee. the average change is an increase in 1.3 stars with a standard devation of .9 stars.(a) is this data paired or not paired? (b) what are the null and alternative hypothesis being tested? (c) what is the t-statistic you would use? write the general formula and the final calculation.(d) sketch a t-distribution, label your t-statistic and the probability you would be interested in: