Hello everyone,
yesterday, while solving Problem 34 (sect. 3.1), I claimed that it is enough to check the normality of a subgroup on generators of the group and of the subgroup and one of you asked me reference for this result. Since I had no time to give a proper answer and Dummit-Foote mentions it without much explanation (between Thm 6 and Prop 7, sect. 3.1) let me inform you that you can find a detailed explanation in this MathStackExchange post: https://math.stackexchange.com/questions/4073741/check-if-subgroup-is-normal-based-on-generators. The post also clarifies a subtlety that I was not careful enough to mention, namely that one has to check that yNy^(-1) is contained in N for all generators y and for their inverses. The solution of the problem does not really change (since rotations commute with each other and the inverse of a reflection s is s itself) but in general one has to be careful.
Thanks for the question that made me check the details and point this out.
Best,
Edoardo