The quaternion group Q may be defined by generators and relations as follows:

Q= <x,y | x^4 =1,y^2=x^2,yxy^-1= x^-1>

for another description let i= squareroot (-1) and take subgroup of 2x2 matrix generated by x= martix going across- 0,-1,1,0 and y = i,0,o,-i

A- explain why every element of Q may be written uniquily in the standard form x^sy^t with s=0,1,2,3 and t= 0,1 hence |Q|= 8

b- verify that the order of y is 4 and xyx^-1 = y^-1

c-complete the following multiplication table representing each result in standard form

x^s (x^s) y

x^s

(x^s)y

d- find all elements of order 2 in Q

e- find the center of Q

please explain as much as you can thanks