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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

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