Find an example infinite group

Question

Find an example of an infinite group and a set of elements $\{g_{1},g_{2},\dots,g_{n},\dots\}$ of finite order with the property that if $$S_n=\cfrac{o(g_{1})+o(g_{2})+\cdots+o(g_{n})}{n}$$ then $\lim_{n\to \infty}S_{n}\in \mathbb{R}-\mathbb{Q}$.

I would appreciate if someone can give me a hint!

Answer 1

Take the infinite sum $\;G:=\bigoplus_{k\in\Bbb N}\Bbb Z/2\Bbb Z\;$ , then for any non-unit elements $\;g_n\in G\;$ we have that $\;o(g_n)=2\;$ , so

$$S_n=\frac{o(g_1)+\ldots+o(g_n)}n=\frac{2+2+\ldots+2}n=2\xrightarrow[n\to\infty]{}2\in\Bbb Q^*$$

Answer 2

There exist groups which contain every element of finite order, for example $$S_{\mathbb{N}}\:\:\text{ and }\:\: \bigoplus_{k\in\mathbb{N}}\mathbb{Z}/k\mathbb{Z}.$$ This means that your question reduces to one about series: find integers $\{i_1, i_2, \ldots, i_n, \ldots\}$ such that if $S_n=\frac{i_1+i_2+\cdots+i_n}{n}$ then $\lim_{n\to \infty}S_{n}\in \mathbb{R}-\mathbb{Q}$. Can you do this?

Note: by $S_{\mathbb{N}}$ I mean all permutations of the integers. The element $(1,2,3,\ldots,n)$ has order $n$.