Let G be a group such that the intersection of all its non-trivial subgroups is a...

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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.) 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.)