Algebra MATH-UA.0343-003
Midterm exam
代数考试代写 Problem 1. Defifine a relation on R123as follows:(x13,y1) ∼ (x2,y2) if and only if there exists λ > 0 such that (x1,y1) = (λx2,y2).
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Problem 1. 代数考试代写
Defifine a relation on R123as follows:
(x13,y1) ∼ (x2,y2) if and only if there exists λ > 0 such that (x1,y1) = (λx2,y2).
Prove that this is an equivalence relation, and describe the equivalence classes.
Problem 2.
Find (with a proof!) the smallest positive integer in the set
A = {24u + 60v + 150w | u, v,w ∈ Z}.
Problem 4.
Let G be an abelian group, and x,y ∈ G are two elements of orders 4. What are the possible orders
of xy ∈ G?
Problem 5. 代数考试代写
Does the set of 2 × 2 matrices
form a group with respect to the matrix multiplication?
Problem 6. 代数考试代写
Let G be the group of rigid plane motions of a square (includes rotations and reflflections). Find
the index of the subgroup generated by the rotation µ (see the picture)