Answer to Question #44731 in Abstract Algebra for jasvinder

Question

Let D(subscript12) = (x,y : x^2 = e ; y^6 = e ; xy =(y^-1) x)
a) Which of the following subsets are subgroups of D(subscript12) ? Justify your answer.
i) (x,y,xy,y^2,y^3,e) ii) (xy,xy^2,y^2^e) iii) (x,y^3,xy^3,e)
b) Find the order of y^2. Is the group (y^2) normal? Justify your answer.
c) Let D(subscript2n) = ( x,y|x^2 = e, y^n = e, xy = xy^(−1) i
Prove the relation
( x^iy^(j+l) if k is even
x^iy^jx^ky^l =
( x^(i+k)y^(l−j) if k is odd.
Further, find all the elements of order 2 in D(subscript12).

c) Find two different Sylow 2-subgroups of D(subscript12).