If you find any mistakes, please make a comment! Thank you.

Exhibit an infinite subset in a group which is multiplicatively closed but not a subgroup


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.

Linearity

This website is supposed to help you study Linear Algebras. Please only read these solutions after thinking about the problems carefully. Do not just copy these solutions.
Close Menu