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In an article, I'm introducing some concepts to give details about them later. For example I have this sentence.

We present our solutions in three groups; theory, implementation, and execution models. In ??? theory group,

What is a correct article for ??? from a',the', or no article, and why? Or is it OK to use the' withfirst/second/third', i.e., in the first theory group,.

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    You might consider "We are presenting our...", though your version is not incorrect.
    – user3169
    Jul 16 '16 at 20:58
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The correct article would be 'the', because you are referring to a single, particular concept that you have already introduced. As a simple example:

"I need a car"

Would mean that you need a car, but doesn't meant that you have a particular car in mind.

"I need the car"

Is referring to a specific car. In your case, I would write:

We present our solutions in three groups: theory, implementation, and execution models. In the first group, theory, we find ...

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