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Adding elements from different cyclic groups


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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.

 

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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 by Nehushtan
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