Virtually cyclic groups: Is the finite permutation group virtually cyclic? & Uncountable groups cannot be virtually cyclic.

Question

I can't answer two questions about Virtually Cyclic Groups. Could anybody help me? I have already proven that the ininite dihedral group is virtually cyclic. I find the questions difficult because I can't think of counter examples or how to construct a subset that works.

A group is called virtually cyclic if there exists a cyclic subgroup of finite index.

1. Prove or disprove: $Sx$ (the group of permutations of a set $X$ with $f(x)\neq x$ for only finitely many $x\in X$) is virtually cyclic.

2. Show that an uncountable group cannot be virtually cyclic.

Answer 1

For the first one:

No (if $X$ is infinite). Suppose $S_X$ were virtually cyclic, with $<g>$ a cyclic group of finite index. But then $G=<g>$ fixes all elements in $X$ outside of a finite set. Since $X$ is infinite (by assumption) there is a countable set in $X^G$, $\{x_i\}_{i=1}^{\infty}$. But then no two of the transpositions $(x_{2i},x_{2i-1})$ define the same coset.

For the second: just cardinality. The cosets of $G$ cover group in question. $G$ is countable and a finite union of countable sets is countable.

Answer 2

Hint:

1. Use 2. and a sufficiently large $X$, for instance.
2. What is the largest possible cardinality of a cyclic (sub)group?