How can $ \langle -1 \rangle $ be the same as $ \text{C2} =\{-1, 1\} $?
Why can we just write $ \langle -1 \rangle $? Why do we say that $ \{-1, 1 \} $ is generated by $ \langle -1 \rangle $? With $ \langle -1 \rangle $ we get $ \{ \ldots, -3, -2, -1 \} $, and with $ \langle -1 \rangle $ we get $ \{1, 2, 3, \ldots \} $. I am really confused.
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How can $\langle -1 \rangle$ be the same as $\text{C2} = \{ 1, -1 \} $?
$ \langle g \rangle $ is defined as the group "generated" by some element $ g $. In particular, we say that a group, $ G $, is generated by $ g $, if $ \langle g \rangle = \{ n \in \mathbb{Z} : g^n \} = G$.
Now, we take $ -1 \in Z^\times$. Now, $ \{ n \in \mathbb{Z} : {(-1)}^n \} = \{ 1, -1 \} $.
Why can we just write $\langle -1 \rangle$ ?
Because they are the equal.
Why do we say that $ \{ -1, 1 \} $ is generated by $ \langle -1 \rangle $?
Well, it is "generated" in the sense, that you can take the set $ \{ -1 \} $ and "extend" it under multiplication and inverses, and you get $ \{ -1, 1 \} $.
Intuitively, $ \langle g \rangle $ can be thought of as the smallest group, with $ g \in G $.
With $ \langle -1 \rangle $ we get $ \{ \ldots, -3, -2, -1 \} $, and with $ \langle -1 \rangle $ we get $ \{1, 2, 3, \ldots \} $.
The confusion is here that $ \langle -1 \rangle $ is really ambiguous, and requires context to be understood. If you consider $ \mathbb{Z} $ under addition, you generate it from adding $ -1 $, and thus you get $ \mathbb{Z} $. If you consider $ \mathbb{Z} $ under multiplication ($ \mathbb{Z}^\times $), the story is different.
The operation here is multiplication, so $\langle -1 \rangle$ is generated by powers of $-1$, which are $1$ and $-1$.