Here are two examples, both due to Shelah:

1. Say an abelian group has a norm if, well, it has a function which behaves like a norm. We say this norm is discrete if the range of this norm in the real numbers is discrete. Clearly every free abelian group has a discrete norm. What about the other direction? It turns out that indeed, if. $G$ has a discrete norm, it is a free abelian group.

2. Let $G$ be an uncountable group, does it have an uncountable subgroup other than itself? If $G$ is abelian, no. But it turns out that there is a non-abelian example.

Here are two examples:

1. Say an abelian group has a norm if, well, it has a function which behaves like a norm. We say this norm is discrete if the range of this norm in the real numbers is discrete. Clearly every free abelian group has a discrete norm. What about the other direction? It turns out that indeed, if. $G$ has a discrete norm, it is a free abelian group. This was shown by Juris Steprāns in

   Steprāns, Juris, A characterization of free Abelian groups, Proc. Am. Math. Soc. 93, 347-349 (1985). ZBL0566.20037.

2. Let $G$ be an uncountable group, does it have an uncountable subgroup other than itself? If $G$ is abelian, yes. But it turns out that there is a non-abelian example of an uncountable group such that every proper subgroup is countable. This was shown by Saharon Shelah in

   Shelah, Saharon, On a problem of Kurosh, Jonsson groups, and applications, Word problems II, Stud. Logic Found. Math. Vol. 95, 373-394 (1980). ZBL0438.20025.