Bourbaki, to be read in any order. The structure of the books makes prior knowledge irrelevant, and they clean past misconceptions and handwaving out of the mind. They are very dense and challenging, though. The first chapter of Algebra can take a while to sink in, but the books are, in spite of protests, independent of the book on set theory. However, after a definition has been read, illustrations will make sense of it, and the illustrations will themselves make sense, because the difficulty comes from having to realize that the definition of the thing is exactly what it takes for something to be that thing, and that the definition is so open that you can make things that satisfy it at will.

After getting used to Algebra, Rotman - "An introduction to the theory of groups" is number theory-minded, and its historical expositions and diagrams add depth and explain, to the same degree Bourbaki doesn't, what it is you're doing. Note however, that neither of these introduce Cayley graphs early, which if she's geometrically minded you'll want to show her how to make as soon as she has a handle on the ideas (you wouldn't want her to rely on them entirely, as they depend on the group axioms).

These lay considerable conceptual groundwork: Between Bourbaki's Algebra and Rotman, she would have enough familiarity with axiomatic constructions and the interest in solutions to polynomial equations that you could introduce the complex numbers axiomatically, by showing how the algebra of the reals adjoined with a token solution to $x^2+1$ is defined completely just by assuming certain axioms on multiplication and addition that apply to the real numbers also apply to the new algebra, the ring axioms. I think that does its job to such satisfaction that it justifies the whole language. It would also be a shortcut from the early chapters of Rotman to the idea of the Galois group, which comes up somewhat late in the book.

Bourbaki's Algebra and Set Theory. The structure of the books makes knowledge outside the series irrelevant, and they clean past misconceptions and handwaving out of the mind. They are very dense and challenging, though. While the books are not considered good introductions by many, I personally used them to start learning mathematics, and while under the same teaching circumstances as the OP, having the student read Algebra was the only way I was able to teach any real facility, in spite of many exercizes. The first chapter of Algebra can take a while to sink in, but in spite of protests it does not require reading Set Theory first, only knowledge of basic set operations and notation and of the product of sets. However, after a definition has been read, illustrations will make sense of it, and the illustrations will themselves make sense, because the difficulty comes from having to realize that the definition of the thing is exactly what it takes for something to be that thing, and that the definition is so open that you can make things that satisfy it at will.

After getting used to Algebra, Rotman - "An introduction to the theory of groups" is number theory-minded, and its historical expositions and diagrams add depth and explain, to the same degree Bourbaki doesn't, what it is you're doing. Note however, that neither of these introduce Cayley graphs early, which if she's geometrically minded you'll want to show her how to make as soon as she has a handle on the ideas (you wouldn't want her to rely on them entirely, as they depend on the group axioms).

These lay considerable conceptual groundwork: Between Bourbaki's Algebra and Rotman, she would have enough familiarity with axiomatic constructions and the interest in solutions to polynomial equations that you could introduce the complex numbers axiomatically, by showing how the algebra of the reals adjoined with a token solution to $x^2+1$ is defined completely just by assuming certain axioms on multiplication and addition that apply to the real numbers also apply to the new algebra, the ring axioms. I think that does its job to such satisfaction that it justifies the whole language. It would also be a shortcut from the early chapters of Rotman to the idea of the Galois group, which comes up somewhat late in the book.

Bourbaki, to be read in any order. The structure of the books makes prior knowledge irrelevant, and they clean past misconceptions and handwaving out of the mind. They are very dense and challenging, though. The first chapter of Algebra can take a while to sink in, but the books are, in spite of protests, independent of the book on set theory. However, after a definition has been read, illustrations will make sense of it, and the illustrations will themselves make sense, because the difficulty comes from having to realize that the definition of the thing is exactly what it takes for something to be that thing, and that the definition is so open that you can make things that satisfy it at will.

After getting used to Algebra, Rotman - "An introduction to the theory of groups" is number theory-minded, and its historical expositions and diagrams add depth and explain, to the same degree Bourbaki doesn't, what it is you're doing.

Bourbaki, to be read in any order. The structure of the books makes prior knowledge irrelevant, and they clean past misconceptions and handwaving out of the mind. They are very dense and challenging, though. The first chapter of Algebra can take a while to sink in, but the books are, in spite of protests, independent of the book on set theory. However, after a definition has been read, illustrations will make sense of it, and the illustrations will themselves make sense, because the difficulty comes from having to realize that the definition of the thing is exactly what it takes for something to be that thing, and that the definition is so open that you can make things that satisfy it at will.

After getting used to Algebra, Rotman - "An introduction to the theory of groups" is number theory-minded, and its historical expositions and diagrams add depth and explain, to the same degree Bourbaki doesn't, what it is you're doing. Note however, that neither of these introduce Cayley graphs early, which if she's geometrically minded you'll want to show her how to make as soon as she has a handle on the ideas (you wouldn't want her to rely on them entirely, as they depend on the group axioms).

These lay considerable conceptual groundwork: Between Bourbaki's Algebra and Rotman, she would have enough familiarity with axiomatic constructions and the interest in solutions to polynomial equations that you could introduce the complex numbers axiomatically, by showing how the algebra of the reals adjoined with a token solution to $x^2+1$ is defined completely just by assuming certain axioms on multiplication and addition that apply to the real numbers also apply to the new algebra, the ring axioms. I think that does its job to such satisfaction that it justifies the whole language. It would also be a shortcut from the early chapters of Rotman to the idea of the Galois group, which comes up somewhat late in the book.