Reference no: EM13125663
Let A = Z + Z + Z, let x_1 = (1,2,1) and x_2 = (1,5,1), and consider the subgroup B = (x_1, x_2) is a member of G. For the quotient group G = A/B, and write (x,y,z) is a member of G for the coset determined by n element (x,y,z) is a member of A.
a) For the subgroups J_1 = {x,y,0 for | x,y is a member of Z} and J_2 = {(0,0,z) | z is a member of Z} is a member of G. Show that G is now the internal direct sum J_1 + J_2.
b) For the subgroups H_1 = {x,y,x for | x,y is a member of Z} and H_2 = {(0,0,z) | z is a member of Z} is a member of G. Show that G is now the internal direct sum H_1 + H_2.
c) Show that H_2 ~ Z.
d) Show that H_1 ~ Z/3Z.
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