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It's true that the letter is symmetrical. But in all the strings one letter maps onto the next letter. A reflection in this string would either reverse the letter as it moves to the right, map it onto itself without changing position, or map it up or down.

I've come back to this after a few days break, it appears that I missed quite the drama. I shall miss drxx but fully respect Norah's decision. I've solved most of the clues but still don't have a scooby how to enter them. Are all there only real words in the grid? Following an earlier hint would give me a down word starting DB, that can't be right - or is it?

The Set A entries are not real words

Thanks candledave, I'm still confused. This may turn out to be the first non-numbers Listener I don't complete in 10+ years of solving.

The key I think to get going with the gridfill is to identify where the first down clue is in set A. You will then know how acrosses and downs split across the groups and you might then have a stab at those that might be in the top and bottom rows to help start building a grid.

Symmetry helps too.

Using Post 36 you can work out how many across and down clues there are in each set and to which set the first and last across clues belong. If it's any help, the first entries I made were 'Snags Kashmiri's jacket...' and 'Historic parks...' See my comment (Post 78) to clarify the instruction regarding symmetry.

Thanks both. Perhaps a large G+T will help, I soldier on....

This is how I interpret what the strings represent and how they might be treated. The titular entry gives the name of the mathematical group we're dealing with. In the completed grid, that titular entry is treated differently to the other across answers in Set A. It is those other across answers that provide the strings.

You need to think of those strings as merely a snippet of a pattern that repeats/recurs ad infinitum to the left and right. The question is how might that infinite recurring pattern be reproduced using various transpositions. All of them could be reproduced if you simply shunted the string x number of letters to the left or right (a translation). But we're not interested in that. The corrected letters in the Set B clues tell you what kind of transpositions we're interested in.

We're also not interested in the trivial - ie rotation through 360 degrees will get you back where you started, as will 2 horizontal reflections or 2 vertical reflections. What we want to know is how you might reproduce that same infinite recurring pattern using non-trivial transpositions. Following transposition, the pattern may be above or below where the original pattern was (following horizontal reflection) or moved along left or right (following vertical reflection), but it's essentially the same infinite recurring pattern. So the question is whether you can reproduce the Set A across strings (other than the titular entry) as part of a recurring pattern following one or more of the (non-trivial) transpositions given by the Set B corrected letters. I gave an answer re string 2 in an earlier post.

In terms of what you have to sketch above the grid, it needs to be of a design given by the 8 unchecked letters in the border of the grid. There are different kinds of such designs possible (try an image search!), but we need one which "corresponds to the highlighted string". This is where I find the preamble unclear. I am reading that to mean that it is a design that meets the criteria given by the Set B corrected letters, as opposed to one which somehow "looks like" the letter string that is highlighted. But I'm starting to doubt whether what I came up with is what they're looking for. Will be interesting to see how this is marked.

Finally got there. Not sure about the mathsy bit yet, but filled the grid. Very clever.

Well, this one has certainly been a workout. I found it tough, and have a masters in maths with group theory as a specialty :-) Mind you, it was some of the clueing I found challenging (but enjoyable), and my familiarity with the theme is perhaps the only reason I was able to complete this one at all.

I know I’m a weaker cold-solver than many here, and that made it tricky to get going with this one. With just five As and eight Bs I ground to a halt and decided to bring in mechanical aid: I wrote some code to try all the possible arrangements of A and B clues (there’s only a few hundred) and see if any of them fitted with the answers I’d got. Dozens did! So I ploughed on, adding each new answer I got to my code until suddenly there were just a few arrangements that matched and they all put a couple of answers in the same place: I could start grid fill, and it went much better from there because I could slot in all my answers and get some hints for the remaining clues.

For anyone still struggling to start grid fill, it was “Employees” with the titular that cracked it for me, but then the fact I’d got the titular at this stage at all was probably because I’d guessed what was going on thematically right at the start so that may not work for everybody…!

I agree with a few comments about the preamble. iratus@74 is spot on: the titular entry is NOT the name of a group, it’s the name of a family of groups, and the strings do NOT represent elements of a group, they represent the particular groups in the family. Also, I agree with cockie@67 that “positioned symmetrically” doesn’t tell us which symmetries apply, and I find that strangely vague for a puzzle that is all about symmetries! If it’s intentionally vague, perhaps it could have said “positioned symmetrically in a manner to be determined”, so that we knew the vagueness was intentional. Finally, the issue touched on by smellyharry@94 is also valid I think: the preamble says that the corrections describe “one of the strings”, but that’s not quite right — the corrections describe one of the groups that are represented by the strings, and we are to highlight the corresponding string. These are all small points, but may have added in small ways to the impenetrability for some solvers, which would be a shame (though a till not an excuse for intemperate language though, in my book, nor for ignoring an admin warning).

I think the construction is otherwise brilliant, and very neat, and the fact we actually had to use mathematical thinking and not just look it up was ingenious. Not to everyone’s taste, perhaps, but no theme is. One of the things I value in Listener crosswords is being introduced to themes I don’t know, or even don’t like, and having my mind broadened a little, and exercised a little, and boy this did that!

I have, as usual, a couple of parsings I can’t quite figure out. Both in Set B, and m each case I see the definition and know the correction and am happy with my answer but cannot figure out the wordplay. “Iced palm on reflecting” and “Glazed top Glaswegian knocked”. My entry is in the post now, but if anyone can just nudge me to see what is going on with these it will settle my mind and enable me to belatedly move on to this week’s puzzle with a clean slate!