Any subgroup of a cyclic group is cyclic

Question

I'm trying to do Exercise 2.2.4 from textbook Groups, Matrices, and Vector Spaces - A Group Theoretic Approach to Linear Algebra by James B. Carrell.

Let $G$ be a cyclic group. Then any subgroup $H$ of $G$ is cyclic.

Could you please verify if my attempt is fine or contains errors? Thank you so much for your help!

My attempt:

Assume $G = \{g^n \mid n \in \mathbb Z\}$ and $H = \{\ldots,g^{-n_2}, g^{-n_1}, g^0, g^{n_1}, g^{n_2}, \ldots\}$ where $0 <n_1<n_2<\cdots<\infty$. We have $n_p$ is divisible by $n_1$. If not, $n_p = q n_1 + r$ where $0 < r < n_1$. Then $g^{r} = g^{n_p - qn_1} = g^{n_p} (g^{-n_1})^q \in H$, which is a contradiction. As such $H$ is generated by $g^{n_1}$.

Many thanks for your verification @freakish :)) – Akira Jun 26 '20 at 10:24 — Akira, Jun 26 '20 at 10:24
Compare alos with this duplicate. – Dietrich Burde Jun 26 '20 at 11:29 — Dietrich Burde, Jun 26 '20 at 11:29

Akira · Accepted Answer · 2020-07-19T05:41:15.580

0

Here is @freakish's comment that answers my question. I post it here to remove this question from unanswered list.

Looks ok to me.

edited Jul 19 '20 at 05:41

answered Jun 26 '20 at 10:23

Akira

17,367

Any subgroup of a cyclic group is cyclic

1 Answers1