Counterexamples to the Finite Order Group Conjecture

In the realm of group theory, one may encounter statements that initially seem plausible but are, in fact, false. One such statement is the claim that if every element in a group G is of finite order, then G itself must be of finite order. However, we will explore why this is not true and provide specific counterexamples to demonstrate the validity of this assertion.

Counterexample with Direct Sum

Consider the group G bigoplusi in mathbb{N} mathbb{Z}/2mathbb{Z}

This group is the direct sum of countably infinite copies of the cyclic group of order 2. Each element in G can be represented as a finite tuple of 0s and 1s, where each entry is either 0 or 1. Since each element is a finite tuple, every element in G has a finite order, specifically either 1 or 2. However, the group itself is infinite because there are infinitely many possible combinations of these entries. This demonstrates that it is indeed possible for a group where every element has finite order to be infinite.

Counterexample with Complex Roots of Unity

A related counterexample involves the multiplicative group of all complex n-th roots of unity, denoted as μn. Each n-th root of unity has finite order dividing n. The group of all such n-th roots of unity for all n (i.e., the union of all μn) is an infinite group. This is because, for any given n, there are infinitely many n-th roots of unity. Therefore, the group is infinite, even though every individual element has finite order.

Historical Context and Open Problems

The question of whether every finitely generated group with all elements of finite order must be finite leads us to the Burnside Conjecture, proposed in 1902. This conjecture, now known to be false, asked whether every finitely generated group with only elements of finite order is finite. The conjecture was proven false, with the construction of infinite finitely generated groups of finite exponent by Golod and Shafarevich.

The Burnside problem with a bounded exponent is particularly interesting. For a given exponent n, we investigate whether every finitely generated group with every element of order dividing n is finite.

Exponent 4 is a notable case, where any finitely generated group is finite. This result was proven by Sanov in 1940.

Prime Exponents (i.e., n is a prime number) are also well-studied. Kostrikin proved in 1958 that any finitely generated group with every element of order dividing a prime p is finite.

For arbitrary exponents, there are specific results. If the exponent is sufficiently large, Zelmanov proved in 1989 that the group is infinite. This leaves the case of small exponents, where the problem remains open. Notably, for exponent 5, the Burnside problem is still unresolved.

Odd Exponents Greater Than 4381 were shown to have infinite finitely generated groups by Adian and Novikov in 1968. This was later refined by Adian in 1995, with a reduction to exponents greater than 101. These results provide a detailed landscape of the possible behaviors of groups under different conditions.

Even Exponents present a more challenging problem. Building on the work of Adian and Novikov, Ivanov showed in 1994 that there are infinite finitely generated groups with exponent n for very large n with specific conditions. These results were later improved to a substantially lower bound, with n large and divisible by 2^9.

These examples and results highlight the complexity and depth of the Burnside problem and its various exponents. The study of these groups continues to be an active area of research in group theory.