Let G be a group such that the intersection of all its non-trivial subgroups is a nontrivial subgroup. Then every element of G has finite order. Theorem 5. Let G be a group (finite or infinite) and H a finite subset of G. TFAE: (1) H is a subgroup of G. (2) Va, b H ab H. Give an example to show the Theorem 5 is false if H is infinite. Theorem 6. For the group G, let Zg = {ge & Vhe G, gh=hg}. Then Za G. (Zo is called the center of group G.)