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Algebra Level pending

Let \( G \) be a set and assume \( (G,\star) \) is a group. Denote by \( e \) the identity element of the group.

Which of the following facts are true? Write your answer in binary notation, which is to be read from left to right, where "1" means "true" and "0" wrong. Thus, if you think the statements 1,2,3 to be true and 4 wrong, write : "1110" .


1) If \( G \subseteq \mathbb R \) and group law is usual multiplication, then \( e=1 \).

2) If \( |G| =4 \), then \( G \) is abelian.

3) If \( G = \mathbb Z / n \mathbb Z \setminus \{ 0 \} \), then n is prime.

4) If \( |G| \) is prime, then \( G \) is cyclic.

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