7. Let A be the abelian group generated by X,Y, Z subject to the relations: 30 X + 24 Y +18 Z=0 21 X +18 Y +15 Z = 0 6X + 21 Y + 24 Z=0. a) Calculate the determinant of the matrix of coefficients of X, Y, Z. What does this tell you about the rank of the abelian group A and the size of the torsion subgroup T(A)? Can T(A) be identified exactly, as a sum of cyclic groups, from this information? If not, list or otherwise describe the possibilities. (Beware: the rank of an abelian group is not the same as the rank of a matrix that defines the relations for that group.) b) Use row and column operations to determine, or verify, the decomposition of T(A) into a direct sum of cyclic groups. c) The map : A + A such that y(x) = 3.x is a homomorphism. Let A[3] and 3A denote respectively the kernel and image of y. Express the subgroups 3A and A[3] as direct sums of cyclic groups. What is the significance of the number 34A[3] ? 7. Let A be the abelian group generated by X,Y, Z subject to the relations: 30 X + 24 Y +18 Z=0 21 X +18 Y +15 Z = 0 6X + 21 Y + 24 Z=0. a) Calculate the determinant of the matrix of coefficients of X, Y, Z. What does this tell you about the rank of the abelian group A and the size of the torsion subgroup T(A)? Can T(A) be identified exactly, as a sum of cyclic groups, from this information? If not, list or otherwise describe the possibilities. (Beware: the rank of an abelian group is not the same as the rank of a matrix that defines the relations for that group.) b) Use row and column operations to determine, or verify, the decomposition of T(A) into a direct sum of cyclic groups. c) The map : A + A such that y(x) = 3.x is a homomorphism. Let A[3] and 3A denote respectively the kernel and image of y. Express the subgroups 3A and A[3] as direct sums of cyclic groups. What is the significance of the number 34A[3]