Consider the group of quaternions Q = {1,-1,i,j,k,-i,-j,-k} a) Prove that Z(Q) = <-1> b) Build Q/Z(Q), listing its elements as cosets xZ(Q)= xbar, for example jZ(Q) = jbar. c) Prove that Q/Z(Q) ≅ K (Klein 4-group), by constructing the multiplication table of Q/Z(Q) and an isomorphism between K and Q/Z(Q).
I think you need to attempt this yourself first. Also, the definitions of the quaternions and Z(Q) would be beneficial.
Z(Q) is the center of the group, and I know it is {1,-1} but dont know how to prove that. For b), would the elements just be -1(1,-1,i,j,k,-i,-j,-k) = (1,-1,i,j,k,-i,-j,-k)bar? And i think i have the multiplication table for c), its a table of subgroups {1,-1},{i,-i},{j,-j}, and {k,-k}, is that right?
The center of a group Z(G) is the set of elements that commute with every element in G. Show you show that for each g in G, 1*g = g*1. That is true by definition of 1. Then for each g in g, -1*g = g*(-1). That should also be simple enough. Then for each other element, you can show an example where z*g =! g*z. (=! means does not equal).
Q/Z(Q) is the set of all cosets xZ(Q) = x{-1,1}. Now for each x in G, compute what x{-1,1} would be (you know, {-1*x, x}). Some of them will be repeats. Don't list them twice in your final answer.
So the subgroups you wrote for c) is indeed the answer to b), but of course you must show your work. As for the isomorphism in c), remember that is a 1-1, onto, and a homomorphism. A homomorphism satisfies f(x*y) = f(x)$f(y), where * is the multiplication of two elements x and y in K, and $ is the multiplication of two elements in Q/Z(Q).
Awesome, you explained that very clearly thank you.
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