Math Problem Statement
Let n ≥ 2 be an integer, and let G = Z/2Z × Z/nZ, where we use component-wise addition as the group operation. 1. Consider the case n = 6, i.e., G = Z/2Z × Z/6Z. Determine the size of |G| and the order of the element (1, 4) ∈ G. 2. Assume now that g ∈ G and gcd(2, n) > 1. Determine the maximal possible value of ord(g). Use this to prove that G cannot be cyclic in this case. 3. Prove that G is cyclic if 2 ∤ n.
Solution
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Group theory
Cyclic groups
Order of elements
Formulas
Order of an element in a direct product group
Theorems
Structure theorem for finite abelian groups
Suitable Grade Level
Advanced undergraduate