Does every Sylow Subgroup have same cardinality?

Question

I know that the result is clear if group is finite. But what if the group is infinite?

Answer 1

Let $P_1$ and $P_2$ be arbitrary finite $p$-groups, and let $G=P_1*P_2$, be their free product. Then $P_1, P_2 \in \operatorname{Syl}_p(G)$(=by definition the set of maximal $p$-subgroups), and hence in an infinite group Sylow $p$-subgroups can have different cardinalities and so, need not to be conjugate.

The proof is not entirely trivial and uses the Kurosh Subgroup Theorem. See also D.J.S. Robinson, A Course in the Theory of Groups, Theorem 14.3.3.