There used to be a "permutations-groups" tag, which I don't see anymore. What happened to it? Can it be put back?
You could check this simply in the list of tags. (When you start to type permu... you are offered the tags with names containing this string. Then you can click on the tag to find more information.) Or you could check the list of tag synonyms. (
Where you can see that the synonym was suggested on March 6 2013 and it was approved relatively recently, on April 20 2016.1)
Maybe it is worth mentioning that tag synonyms are often discussed on meta before they are created. Typically in threads like http://meta.math.stackexchange.com/questions/22348/tag-management-2016 (or older editions of this thread), but if there is a more extensive discussion, there might be a separate question. I do not recall whether this particular tag synonym was discussed on meta, but if it was I was not able to find a post where it was discussed.
Some basic information about tag synonyms can be found here: What are tag synonyms and merged tags? How do they work?
Let me also address the last part of your question: "Can it be put back?"
Only the moderators can undo the tag synonyms. But they will not do so without a good reason. So probably the way to go would be to post a question here on meta and put forward some arguments why you think the two tags should be separate. You can find a bit about this in the section entitled "How can I delete/reverse/undo bad tag synonyms?" in What are tag synonyms and merged tags? How do they work?
I should probably also point out that even if the tag synonym is cancelled, that does not mean that questions which were synonymized as permutation-groups to permutations will get the original tag after the synonym is cancelled. And important think here is whether the tags were only synonymized, or they were also merged. See here for more details: What happens with the tagged questions when a tag-synonym is cancelled?
1 I have removed this after Daniel Fischer's comment. It seems that the meaning of the dates shown in that list is not entirely clear to me.