So I am doing a project and was given the group Z2 x Z9 and was told it was a noncyclic group. However, when finding the subgroups (1,1), (1,2), (1,4), (1,7), and (1,8) all generate the group. Am I doing it wrong or is the group actually cyclic?
2026-04-02 06:29:54.1775111394
Is the group Z2 x Z9 cyclic?
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It is indeed a cyclic group. In general, $\mathbb{Z}/n_1\mathbb{Z}\times\mathbb{Z}/n_2\mathbb{Z}$ is cyclic if and only if $n_1$ and $n_2$ are relatively prime - this is a good exercise. In particular, if $n_1$ and $n_2$ are relatively prime, then $$\mathbb{Z}/n_1\mathbb{Z}\times\mathbb{Z}/n_2\mathbb{Z}\cong\mathbb{Z}/n_1n_2\mathbb{Z}.$$ And a similar statement is true for products of finitely many cyclic groups.