I just learned incidentally in the comments of another question here that it is not true that every proper subgroup is contained in a maximal (proper) subgroup. A counterexample is easy to find: the additive group $\mathbb{Z}[\frac{1}{2}]$ has no maximal proper subgroup. But the confusion with proper ideals which are contained in a maximal (proper) ideal, by Zorn's lemma, is certainly something that will surprise others. (The reason for the difference, as I see it, is that in the case of ideals we can detect properness by $1 \not\in I$, whereas for subgroups there is no such criterion.)
↧