Centralizers and conjugacy. Let G be a group. For any element a EG, the centralizer Ca of a is the set of all elements which commute with a: Ca = {x € G: xa = ax} = {x € G:a= xax1}. (1) Show that Ca is a subgroup of G. (2) Show that xax-1 = yay-1 if and only if xCa =yCa. (3) Show that the conjugates of a are in bijective correspondence with the left cosets of Ca. Thus if G is finite, the conjugacy class [a] has [G: Ca] elements, and the size of the conjugacy class divides the order of the group. (4) Let p be prime and let G be a group of order pa for some a > 1. Recall that the center of G is C(G) = {a EG: ax = xa for all x EG}. Show that C(G) + {e}. (Hint: Use problem

the smallest possible measurement of the width is 14 feet

step-by-step explanation:

let's assume length of parallelogram is 'l'

width of parallelogram is 'w'

the perimeter of a parallelogram must be no less than 40 feet

so, we get

we are given

length of parallelogram is 6 feet

so,

l=6

we can plug it

now, we can solve for w

subtract both sides by 12

divide both sides by 2

and we get

so,

the smallest possible measurement of the width is 14 feet

.632

step-by-step explanation:

6 * 1/10 = 3/5

3*0.01 = 0.03

3/5 + 0.03 = .63

2*0.002 = .002

.63 + 0.002 = .632

answer: a. 13/44

step-by-step explanation: judging by the information given to us, we are looking for experimental probability. we can see that 13 males have chosen chocolate ice cream. there are 44 ice creams in total (remember that the total is always the denominator). so our answer is 13/44.