
Abstract Algebra
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
ccomplete 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

(A)
x has order 4, so are distinct elements; they are also distinct from y (otherwise Q would be cyclic). So we have five distinct elements. And since , it follows that Q has at most eight elements: . All you need to do is to show that the last three of these elements are distinct from the first five and from each other. For example, for xy:
 If , then (contradiction). Hence .
 If , then (contradiction). Hence .
 If , then (contradiction). Hence .
 If , then (contradiction). Hence .
 If , then (contradiction). Hence .
Show similarly that , then show that .
(B)
We have and .
… and the rest is straightforward.
(C)
For any , where and .
(D)
is the only element of order 2. This is just a matter of checking one by one.
(E)
We have ; hence x and y do not belong to the centre, and neither do their inverses and . Also but . Hence, neither xy nor belongs to the centre. That leaves . Simple checking should verify that does commute with every other element in Q. Hence .