The easiest examples are abelian groups, which are direct products of cyclic groups. The Klein V group is the easiest example.

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There is a general way to approach questions of the form "find a non-cyclic gp. (or ab. gp.) all of whose proper subgroups are cyclic", "find a non-ab. gp. all of whose proper sgs. are abelian", etc., namely:

2016年2月15日 · Order of elements in a noncyclic group. Ask Question Asked 9 years, 1 month ago. Modified 9 years, 1 month ...

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One thing you can try is find the groups of each order. A group of order $2$ must be isomorphic to $\mathbb{Z}_2$, which contains identity and another element of order $2$.

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2015年7月8日 · The problem is to prove or disprove that there is a noncyclic abelian group of order $51$. I don't think such a group exists. Here is a brief outline of my proof: Assume for a contradiction that there exists a noncyclic abelian group of order $51$. We know that every element (except the identity) has order $3$ or $17$. Assume that $|a|=3$ and ...

2013年9月23日 · Construct an abelian group of order 12 that is not cyclic. Can somebody please explain me with examples non-cyclic groups I'm having a hard time understanding.