Is it true that every infinite finitely generated group contains an element of infinite period?
I think it is true, but I can't prove it.
Does every infinite finitely generated group contains an element of infinite period?
0
$\begingroup$
abstract-algebra
group-theory
-
0Google "Tarski Monster" . – 2017-01-23
-
0See the answer by "ndp" for a finitely generated counterexample at the duplicate. – 2018-02-13
2 Answers
2
"This was once a paradox but now time gives it proof".
The simplest example is a consequence of the Golod-Shafarevitch theorem:
Theorem:For each prime $p$ there is an infinite group generated by three elements where each element has finite order a power of $p$.
See Herstein, Noncommutative Rings.
The proof of this is understandable and not too difficult. On the other hand there are a number of much more extreme examples (mostly with Russian names), whose proofs are just as monstrous, under the general heading of Burnside's Problem.
5
The Grigorchuk group is finitely generated, and every element has finite order, but it is still infinite.