📂Topological Data Analysis

Definition of Torsion Subgroups

Definition ¹

Let $G$ be an Abelian group.

1. If $g \in G$ satisfies $ng = 0$ for some $n \in \mathbb{N}$, then $g$ is said to have a Finite Order.
2. If the set $T \subset G$ of all elements of $G$ with finite order is a subgroup of $G$, then $T$ is called the Torsion Subgroup of $G$.
3. If the torsion subgroup of $G$, $T$, virtually Vanishes, meaning $T = \left\{ 0 \right\}$, then $G$ is said to be Torsion-free.
4. If $T$ consists of finitely many elements, the cardinality $\left| T \right|$ of $T$ is called the Order of $T$.

Description

Torsion?

Torsion can be translated as ’twisting’ or ‘wringing’, but in [Abstract Algebra](../../categories/Abstract Algebra), it doesn’t really give that intuition, so it’s recommended to just accept ’torsion’ as it is. Personally, regardless of Korean or English, the fact that $T$ is a finite group kind of gives the impression of overturning the whole group $G$, but I see no need to forcibly make the readers understand it.

Torsion Subgroup

$$G_{1} = \mathbb{Z}_{2} \times \mathbb{Z}_{3} \times \mathbb{Z}^{5}$$

Consider, for example, the group $G_{1}$ mentioned above. $_{1}G$ has infinitely many subgroups, but among them, only $T_{1} \subset G_{1}$, which is isomorphic to $\mathbb{Z}_{6} \simeq \mathbb{Z}_{2} \times \mathbb{Z}_{3}$, is a torsion subgroup. According to the definition, it has to be a subset which is not just any finite group but the set of all elements with finite cardinality.

Torsion-free

Another example is $G_{2} = \mathbb{Z}^{5}$. Even if $0$ is the only element with finite order, since $1 \cdot 0 = 0$ for $1 \in \mathbb{N}$, the trivial group $\left\{ 0 \right\}$, which is a torsion subgroup, exists. Therefore, even though one wants to say there is no torsion (torsion-free), one cannot directly say ’none’ and instead uses the term Vanish.