Final answer:
The incomplete question requires a demonstration of properties concerning torsion groups or torsion-free groups, which are concepts from abstract algebra in mathematics.
Step-by-step explanation:
The question seems to be incomplete, but from the provided information, it appears to pertain to the concept of torsion groups and torsion-free groups within the field of abstract algebra.
To prove properties related to these groups, one typically uses group theoretical notions and the properties of group elements, specifically their orders.
A torsion group is a group where every element has a finite order, meaning the element raised to some finite power equals the group's identity. Conversely, a torsion-free group is one where the only element with finite order is the identity element itself.
A group is torsion-free if and only if its only element of finite order is the identity. To prove this, assume torsion-freedom, showing any non-identity element lacks finite order.
Conversely, assume the identity is the sole element of finite order, demonstrating torsion-freeness. In a torsion-free group, raising a non-identity element to any power never yields the identity, affirming that the identity is the only element with finite order. This concise proof captures the fundamental relationship between torsion-free groups and the singularity of finite-order elements within them.