If $G$ s.t. $|G|=15$ has only one subgroup of order $3$ and only one of order $5$, then $G$ is cyclic. Generalize to $|G|=pq$, for $p,q$ primes.

Question

Let $G$ be a group where $o(G)=15$. If $G$ has only one subgroup of order $3$ and only one of order $5$, prove that $G$ is cyclic. Generalize to $o(G)=pq$, where $p$ and $q$ are primes.

For order $pq$ see Herstein's exercise here. – Dietrich Burde Jun 09 '15 at 19:00 — Dietrich Burde, Jun 09 '15 at 19:00

score 0 · Answer 1 · answered Jun 09 '15 at 18:58

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Hint: If $N$ and $M$ are normal subgroups such that $N\cap M=1$, then $nm=mn$ for every $m\in M$ and $n\in N$.

answered Jun 09 '15 at 18:58

Bobby

7,396

Yes, see here. – Dietrich Burde Jun 09 '15 at 18:59

If $G$ s.t. $|G|=15$ has only one subgroup of order $3$ and only one of order $5$, then $G$ is cyclic. Generalize to $|G|=pq$, for $p,q$ primes.

1 Answers1