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I am looking for some examples of non-cyclic abelian groups.

I found something like order 12, or other group.

Here i am looking for simplest (smallest element) non-cylcic abelian groups.

If you know something about this please let me know.

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The smallest noncyclic group is the four element Klein four-group https://en.wikipedia.org/wiki/Klein_four-group . All finite abelian groups are products of cyclic groups. If the factors have orders that are not relatively prime the result won't be cyclic.

It doesn't make sense (in general) to ask for "small elements" of groups.

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The Klein-four-group is an abelian group with 4 elements, $1, a, b, c: a^2 = b^2 = c^2 = 1$ and, $ab = c, ac = b, bc = a$.

Other small non-abelian groups include $D_n$, the dihedral group, and $S_n$, the symmetric group. The order of $D_n$ is $2n$ and $S_n$ is $n!$.

Edit: I misread non-cyclic abelian groups as non-abelian. The family of dihedral groups is a product of abelian groups $Z_2, Z_n$ but the product is not direct (direct product would be abelian), it is a semi-direct product.

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