List questions from direct-product | MathQa

Is there a slick way to test whether $\Bbb Z_{mn}\cong \Bbb Z_m\oplus \Bbb Z_n$?

Published on 12 Jan 2025 - 9:28

Structure of groups of order $pq$, where $p,q$ are distinct primes.

Published on 12 Jan 2025 - 9:28

Prove that $G \times H$ is cyclic $\iff$ $\gcd(m,n) =1$

Published on 12 Jan 2025 - 9:28

Group Theory: let $G$ be a group and let $G=H\times K$, is it true that $G/H\cong K$?

Published on 12 Jan 2025 - 9:28

Proving the direct product $S^3 \times \mathbb R^{+}$ is isomorphic to $H^{*}$

Published on 12 Jan 2025 - 9:28

Which of the following group is isomorphic to the group $S_3\times \mathbb Z_2$?

Published on 12 Jan 2025 - 9:27

Determing the isomorphism class of $U(16)$ and writing it as an internal direct product of cyclic groups.

Published on 12 Jan 2025 - 9:28

How to show $\frac{\mathbb{Z}_m\times \mathbb{Z}_n}{\langle (a,b)\rangle}\simeq \mathbb{Z}_{\frac mc}\times \mathbb{Z}_{\frac nd}$?

Published on 12 Jan 2025 - 9:28

Find the order of $\langle (1,1)\rangle$ in $\mathbb{Z}_2 \times \mathbb{Z}_4$.

Published on 12 Jan 2025 - 9:28

What are the distinct cyclic subgroups of order 12 in $\mathbb{Z}_6 \times \mathbb{Z}_{10}^\times$?

Published on 12 Jan 2025 - 9:27