There is a similar post before. But it did not include my question…..
It is asking whether a subset of a complex number is a subgroup of $\mathbb{C}$ under addition.
Subset is: $\pi^n | n\in \mathbb{Z}$.
I thought it is a subgroup. However, the answer was No. And the reason is "not closed under addition". But, $\pi+\pi^2$ is still in $\mathbb{C}$. And from the textbook, there is a theorem stating that if $G$ is a group and $a \in G$, then
$$H=\{a^n|n\in \mathbb{Z} \}$$
is a subgroup of $G$.
But why the question is "Not a subgroup". I understand that the checking procedures is (1)closure (2)inverse.
Best Answer
$\pi + \pi^2$ is in $\mathbb{C}$ but $\pi + \pi^2$ does not equal $\pi^k$ for any $k \in \mathbb{Z}$ so $+$ is not closed in the subset.
Also:
$$a^n = \underbrace{a \cdot a \cdot \cdots \cdot a}_{n \text{ factors}}$$
Where the dot is the group operation. So in your case, in more standard notation $a^n$ becomes $\underbrace{a + a + \cdots + a}_{n \text{ factors}}$ which is more commonly written $na$.