Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.4
Give an explicit example of a group $G$ and an infinite subset $H$ of $G$ which is closed under the group operation but is not a subgroup.
Solution: There are infinitely many nonnegative integers, and the sum of two nonnegative integers is nonnegative. However, the additive inverse of a positive integer is negative. So the natural numbers $\mathbb{N}$ constitute an infinite subset of the additive group $\mathbb{Z}$ which is closed under the operator but not a subgroup.
