?

Abstract Algebra

Free Version

Upgrade subject to access all content

Moderate

Direct products and properties

ABSALG-UETEKV

Which of the following statements is true about direct products?

A

If $G, H, \text{ and} K$ are abelian groups, such that $G\times H\cong G\times K$, then $H\cong K$.

B

$\mathbb{Z}_2\times \mathbb{Z}_{n}\cong\mathbb{Z}_{2m}$ for all integer $m\geq 2$.

C

The direct product of two groups is an abelian group.

D

The direct product of finite groups is a finite group.

E

$\mathbb{Z}\times\mathbb{Z}$ is not cyclic.