Prove that an Infinite group must have subgroup with infinite elements.
I know that if group was cyclic order of the generator is infinite and there are infinite number of divisors.
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1Do you mean that it must have an infinite number of subgroups or subgroups which are infinite? – DonAntonio Oct 03 '13 at 18:03
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@DonAntonio infinite number of subgroups. Edited the question. – Surya Oct 03 '13 at 18:05
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@Surya I think you should ask this as a separate question, so as not to nullify Trevor Wilson's answer. – user1729 Oct 03 '13 at 18:07
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(But for a hint: Consider the subgroup generated by each element. When do these subgroups precisely coincide?) – user1729 Oct 03 '13 at 18:07
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1@user1729 ok edited it back. – Surya Oct 03 '13 at 18:21
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For the other interpretation of the question, you can check the thread Finite number of subgroups ⇒ finite group. – Jeppe Stig Nielsen Dec 30 '21 at 22:12
2 Answers
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Any group is a subgroup of itself.
Trevor Wilson
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11If we disallow this trivial case, the statement is wrong: see http://math.stackexchange.com/questions/261145/find-an-abelian-infinite-group-such-that-every-proper-subgroup-is-finite – Trevor Wilson Oct 03 '13 at 17:46
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@Surya Do you mean trivial cases? That is what Trevor Wilson saying. The trivial group is finite... – user1729 Oct 03 '13 at 18:05
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There are infinite groups with no infinite proper subgroups. Even if the group is abelian, as show the $p$-component of $\mathbb{Q}/ \mathbb{Z}$ (that is the set of elements whose order is a power of $p$), for any prime number $p$.
Tarski monsters provide examples in the non abelian case.
Yassine Guerboussa
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However, all known examples are infinitely presented. Which is interesting. – user1729 Oct 03 '13 at 18:18
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May be you can edit the question to include your remark. Or better ask the question independently : Is there a finitely presented infinite group in which every proper subgroup is finite? – Yassine Guerboussa Oct 03 '13 at 18:24
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When I said "all known examples" I meant all known examples (not just those known by me). It is a relatively famous open problem, related to Burnside's problem. See the discussion following Problem 1.1 of this list of problems. – user1729 Oct 03 '13 at 18:35
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I remember well, that I found the Riemann Hypothesis as an exercise in Serge Lang's "Complex Analysis". – Yassine Guerboussa Oct 03 '13 at 20:19