In one book it says
a set of permutations with the composition operation is a group. This implies that using two permutations one after another cannot strengthen the security of a cipher, because we can always find a permutation that can do the same job because of the closure property.
If I try to understand the bold part, is below explanation correct?
Assume you have a set of all permutations of "abc", that is: Your set is: "abc" ,"acb", "bca", "bac", "cab, "cba". Let's take "abc".
Permute it once and get say: "cba" (Let's say permuting once means you encrypt it).
Now let's assume you want to strengthen above permutation by permuting it once again (encrypt it once more), e.g. now you permute "cba" and arrive at "bac". In theory one could have arrived at "bac" from "abc" in a single permutation too (in a single encryption), thus additional permutation didn't really make much sense from this point of view. Because it says basically what you can do in two permutations you can effectively also do in a single permutation.