Let A be the abelian group generated by X, Y, Z subject to the relations: 4X - 6Y - 4Z=0 4X + 6 Y - 2 Z=0 12 X +8 Y + 10 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 V: A + A such that "(x) = 2 x is a homomorphism. Let A[2] and 2A denote respectively the kernel and image of 4. Express the subgroups 2A and A[2] as direct sums of cyclic groups. What is the significance of the number (2A| * |A[2]|? Let A be the abelian group generated by X, Y, Z subject to the relations: 4X - 6Y - 4Z=0 4X + 6 Y - 2 Z=0 12 X +8 Y + 10 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 V: A + A such that "(x) = 2 x is a homomorphism. Let A[2] and 2A denote respectively the kernel and image of 4. Express the subgroups 2A and A[2] as direct sums of cyclic groups. What is the significance of the number (2A| * |A[2]|