Bismillahir Rahmanir Rahim.
At first, I want to say that most of us have a wrong notion about the matter of talaq specifically in the mentioned case.
If 1 or 2 talaqs are given:
If anybody gives one talaq or even two talaqs, they still can re-validate their marriage within next three menstrual period (or birth if the girl is pregnant) of the girl by telling sentence like "I am reverting my talaq." or by acting any intimate work that only husband and wife could do.
If three menstrual period of the girl overs (or given birth if the girl is pregnant) and the third talaq is not committed, then they have to remarry by agreeing on a mohr.
If 3 talaqs are given:
But if three talaq is given, then there remains no option to remarry except the girl gets married with another boy and have intercourse then gets divorced.
How to count talaq:
Remember one thing, if anybody give one talaq and remarry/re-validate, then give one talaq and remarry/re-validate and then again gives one talaq, then they could not remarry or re-validate their marriage. It is our wrong notion that if I say "one talaq", then only one talaq occurs, rather if I give one talaq now and I I gave two talaqs before, then it would be counted as three talaq.
So now, come to the point of the solution of the mentioned case.
Are they permitted to remarry after one talaq was given to the woman and them been apart for 5 years?
Well, if they did not do intercourse within the next three menstrual period of the woman calculating from the day of giving talaq (or before given birth if the woman was pregnant) then they have to remarry just determining a mohr only. And they have to repent to the Allah subhanahu wa ta'ala as they did zina in these 5 years.
Otherwise, their marriage is still valid.
However, if they could not remember whatever they did intercourse or not within the time duration mentioned, then it is better to remarry for safety.
And Allah Subhanahu wa ta'ala knows the best.
If you need reference, just google it, you would find a lot. If I had time in hand, then I would give. I found this arbitrary reference by googling just in short time.