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A very smart non-mathematician friend is looking to learn about groups, and I was wondering if people might have suggestions (this is NOT a duplicate of this question, since a textbook is not what I am looking for, at least not at first).

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Have a look at the Numberphile video series on youtube. They have a lot of videos for non-mathematicians who are interested! – Ed_4434 Mar 28 at 13:46
Your non-mathematician friend might find the book Abel's Proof, by Peter Pesic, worth a look. – Barry Cipra Mar 28 at 13:56
Why exactly do they want to learn about groups? If it is for an application (e.g. point group theory in inorganic chemistry) then it might be best to go for a presentation that is interlaced with the application, rather than a standard mathematical presentation. – Ian Mar 28 at 14:00
@Ian My friend is interested on general principles (she is actually a film-maker!) – Igor Rivin Mar 28 at 14:51
If you round to the nearest hour, people use an Abelian additive group of order $12$ or $24$ every day when working with time. That's secretly everyone's first group. – Todd Wilcox Mar 28 at 18:47

Groups and Symmetry: A Guide to Discovering Mathematics, by David W. Farmer.

The highlighted title may convince that it assumes not too much mathematics for the learner. It is very little book, not of the type Definition-Theorem-Proof.

At least (in on-line preview) I don't find a single concrete mathematical statement, but always beautiful pictures and only that.

enter image description here

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I recommend the Marcus du Sautoy's book: Finding Moonshine : Mathematician's Journey Through Symmetry
enter image description here

He is also the author of The Music of the Primes.

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Read this. Don't think the author knew how to dumb it down without really dumbing it down. – djechlin Mar 29 at 2:56

Try The Fascination of Groups by Budden.

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Two books not yet mentioned that the OP might want to consider are:

Groups in the New Mathematics by Irving Adler (1967)

Groups and Their Graphs by Israel Grossman and Wilhelm Magnus (1964)

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I recommend Carter's Visual Group Theory. It makes heavy use of pictures and diagrams (hence the name) and I found it very clear.

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I would suggest to read, Simon Singh's Fermat's Last Theorem.

It starts with the introduction to simple problem, solutions. And entire history as it evolves to solve the problem surrounding group theory. And then into advanced theory.

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