Exercise 981Algebra › GroupsL1Added 3 Nov 2026

Direct products of groups

#direct product#product group#order

Statement

Let $G$ and $H$ be groups. Show that $G\times H$ is a group under $(g,h)(g',h')=(gg',hh')$ , and compute $|\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}|$ .

Hint

The group axioms hold component-wise.

Show solutionHide solution

Solution

Group axioms. Closure, associativity, identity $(e_G,e_H)$ , inverse $(g^{-1},h^{-1})$ all hold component-wise.

Order. $|G\times H|=|G|\cdot|H|$ , so $|\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}|=6$ .

Isomorphism. $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}\cong\mathbb{Z}/6\mathbb{Z}$ (Chinese remainder theorem, since $\gcd(2,3)=1$ ).

Conclusion. The direct product is a group of order $|G||H|$ .

Direct products of groups

Statement

Solution

More problems in Groups