Abstract Algebra Answers
Abstract Algebra – Groups ?
i) Let G be an abelian group and
Cube (G) = {g^3 I g is an element of G}
Is Cube (G) a subgroup of G? Prove your answer.
ii) Let GL2(Z) (Z represents the set of integers) be the set of all 2X2 matrices with entries in Z (The set of all integers) and non-zero determinant. Is GL2(Z) (Z represents the set of integers) a group under matrix multiplication? Prove your answer.
i) Well, let g, h be elements of G, so that g^3, h^3 are elements of Cube(G). We want to show that g^3(h^3)^(-1) is an element of Cube(G). We know that (gh^(-1))^3 is an element of Cube(G). Are these two the same?
ii) This question is a little weird, because typically GL2(Z) means 2×2 matrices with entries in Z and with determinant ±1. Under this assumption, then yes, GL2(Z) is a group.
However, under your definition, it is not. Recall that det(AB) = det(A)det(B) for all matrices. So if A and B are inverse to each other, then det(A)det(B) = det(I) = 1. If, say, det(A) = 2, then det(B) = 1/2, which can’t happen if B has entries in Z. So… can you come up with a 2×2 integer matrix of determinant > 1?
Basic abstract algebra, pt.1
