Grouping a set of numbers

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I have a set of numbers which I don´t know if they belong to the same group (I could also call it factor or treatment, but actually each group is suppose to identify the same biological event).

I am assuming that this set can contain numbers belonging to 1 or more groups. I am going to compute several sets of numbers and for the majority of them I would expect just 1 group. Moreover, in the majority of the sets which have more than 1 group they are expected to have no more than 5 groups.

I am aware that this question can be very cloudy, however, I`am looking for some advice to at least read more about some specific mathematical topic that could help me in the purpose.

Let´s try to give a very simplified example. Imaginary set 1:

set1 <- c(2000, 2500, 2400, 3600, 1500) 

Imaginary set 2:

set2 <- c(1000, 1400, 3800, 1200, 3700)

Here I could expect something like 3 groups to set 1: group 1 : 2000, 2500, 2400 and group 2: 3600 and group 3: 1500.

To the set 2 I would expect two groups: group 1 : 1000, 1400, 1200 and group 2 : 3800, 3700.