Order of element in external direct product

Dependencies:

1. External direct product is a group

Let $G$ and $H$ be groups. Let $(g, h) \in G \times H$. Then $\operatorname{order}_{G \times H}((g, h)) = \operatorname{lcm}(\operatorname{order}_G(g), \operatorname{order}_H(h))$.

Proof

Let

• $o = \operatorname{order}_{G \times H}((g, h))$
• $o_g = \operatorname{order}_G(g)$
• $o_h = \operatorname{order}_H(h)$
• $l = \operatorname{lcm}(o_g, o_h) = o_gk_g = o_hk_h$
• $M(x, y) = \{z: x \mid z \wedge y \mid z\}$. So $\operatorname{lcm}(x, y) = \min M(x, y)$.

\begin{align} & (e_G, e_H) = (g, h)^o = (g^o, h^o) \\ &\implies g^o = e_G \wedge h^o = e_H \\ &\implies o_g \mid o \wedge o_h \mid o \\ &\implies o \in M(o_g, o_h) \\ &\implies o \ge \min M(o_g, o_h) = l \end{align}

$$ (g, h)^l = (g^l, h^l) = (g^{o_gk_g}, h^{o_hk_h}) = (e_G, e_H) \implies o \mid l \implies o \le l $$

Therefore, $l = o \implies \operatorname{order}_{G \times H}((g, h)) = \operatorname{lcm}(\operatorname{order}_G(g), \operatorname{order}_H(h))$.

Dependency for:

1. Zm × Zn is isomorphic to Zmn iff m and n are coprime

Info:

• Depth: 2
• Number of transitive dependencies: 2

Transitive dependencies:

1. Group
2. External direct product is a group