I'm reading a paper currently that uses three groups; the cyclic groups of order 3, 5, and 11. It represents an element from C₃ as (c, 0, 0) an element from C₅ as (0, c, 0) and an element from C₁₁ as (0, 0, c).

What is the motivation for representing the elements like this? i can understand that it is easier for calculations but if someone asked me why you can represent each element as a vector i would not be able to explain why you are allowed to that.

 

The group being considered is most likely [latex]C_3\times C_5\times C_{11}[/latex] (or a group that contains this group as a subgroup). An element in the group is not a vector; it’s just represented by an ordered triple.

The subgroups [latex]\{(c,0,0):c\in C_3\}[/latex], [latex]\{(0,c,0):c\in C_5\}[/latex] and [latex]\{(0,0,c):c\in C_{11}\}[/latex] are isomorphic to [latex]C_3[/latex], [latex]C_5[/latex] and [latex]C_{11}[/latex] respectively, so the individual cyclic groups can be thought of as being embedded in the group in question.

Edited March 21, 201312 yr by Nehushtan