Question #105105
Check whether or not R is a normal subgroup of the group H = (R + Ri + Rj+ Rk,+ )where i²=j²=k² =-1,ij=-ji,jk=-kj,ki=-ik=and
( a+ bi + cj+ dk) + (a ′ + b′i + ′jc + d′ )k = (a + a′) + ( b+ b′ i) + (c + c′ j) + ( d+ d′ k) for
d,c,b,a,d',c',b',a ′∈ R.
1
Expert's answer
2020-03-17T12:25:09-0400

Clearly,0R0\in\R .let a,bRa,b\in\R then a+(b)=abRa+(-b)=a-b\in\R .

Hence ,by one step subgroup test R\R is a subgroup of HH

Let rr be any element of H.H.

Then, r+R=R+r , rHr+\R=\R +r \space, \forall \space r \in H

Since,a+bi+cj+dk+x=a+bi+cj+dk+xa+bi+cj+dk+x=a'+b'i+c'j+d'k+x  a,b,c,d\forall \space a,b,c,d a,b,c,d,xRa',b',c',d',x\in\R .

Hence ,R\R is a normal subgroup of HH by the definition of normal subgroup



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