Stack Exchange's search function is notoriously bad. Try Google, Bing, etc.
If you searched diligently, and you did a good job of writing your question in a general form, and your question turns out to be a duplicate, that is not a terrible outcome. It means that you will help others find the answer via search, because now anyone else who searches on "finite group criterion" instead of "finite group test" will readily find your post and then be able to follow the link to the answer.
Check a textbook. Find the concept in a textbook. Check what phrasing they use to describe it, then search using that phrasing. Also, sometimes you will find the answer in the textbook, and if not, it provides some context you can use to improve the question.
Sometimes, it is helpful to use subsets of the words when searching.
When you write a post, make sure to be clear about what you are asking. We are a question-and-answer site, so we require you to articulate a specific question. A question usually ends with "?". Don't force us to guess what your question is. In your post, I can't tell what your question is. Your post contains only statements, but no question. You mark a section as "Question:" but then what follows it is a declarative statement, not a question. So I find it painful to try to imagine what you might be trying to ask.
Also, if you are going to ask about "the converse", and it is not 100% clear what you mean by that, I suggest that you state explicitly the result you are trying to prove. Textbooks always show a careful statement of the theorem before trying to present a proof. Try to do the same. You need to be very clear about what you are trying to prove, before you can have any hope of proving it. Help us out, by providing all of the parts you can do yourself.