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greengage - if my understanding is correct, the top left string fails the second thing the Set B message tells you. I've highlighted something in the bottom right.

As a few others have said, extremely difficult to get started this week, but got significantly easier once you had some entered into the grid. Impressive stuff, Hawk!

Thanks mondo, will ponder!

Thanks mondo, will ponder. Looking at the sketch, the lower right letters are discernible to the eye of faith!

There's a different theme to the crossword every week: some of them I know something about, others I have to look up, often learning something worth knowing. The quality of a crossword is independent of how interested I am in the subject. "This is not my bag" is fair comment, "this is dross because I don't like the theme" is not.

I do think that we owe setters some consideration in our comments.

Having said that, whereas the groups used this week are not my area of expertise, from what I've read I think the setter has got the nomenclature wrong. The strings are not elements of one mathematical group, they're each an example of a different group, the groups being of a related kind. Which misled me slightly early on when I was trying to guess what was going on, but didn't stop me enjoying the puzzle.

cockie - isn’t the symmetry point left slightly vague thematically.

Unless I’ve gone wrong I wouldn’t necessarily call the end result S symmetry.

I would just like to record my thanks to drxx for the enormous amount of time and effort that he has dedicated to helping struggling solvers to make progress with the Listener puzzles.

This of course does not diminish the debt of gratitude owed to other regular posters who offer help, and to Norah for making it possible.

I've just joined this forum but have been doing the Listener for about a year and benefiting from some helpful posts. Thank you. I enjoy all the puzzles, because they have clearly been set by clever people, provide varied challenges and cover all sorts of themes.
Some of the posts about maths surprise me. There aren't many mathematical puzzles and the maths involved is never more than is taught for GCSE. The puzzle about perfect numbers last month was a brilliant piece of logic. I'm a linguist by the way.
Regarding this puzzle, like Mondo says, it was hard to get going but once I got the last across and down clues, it got easier.
By the way, symmetry is either rotational (like the letter S) or line symmetry (like the letters A or B). This puzzle is both (like the letters X and H), so apart from not wanting to give the game away too much, there was no need to specify the type of symmetry.
I think Greengage is right: the string top left does meet the two parts of the instruction.
Can anyone give me a guide on the string top right? I have the last three letters but my answer for 'big house at Kelso' does not provide a first letter that works.

The string top left fails the first part of the description.

For the top-right string, the first letter appears in two other strings also.

Thanks, Iratus. I must be missing something in the instruction. My first letter top left maps as instructed onto the second letter, the pair are then repeated.
One more thing: I can't identify the misprint in the Dicky Lane clue. I think it's in the 6th word but I can't find an adjusted synonym for the answer.