Tagged: direct product of groups
Prove that a Group of Order 217 is Cyclic and Find the Number of Generators
Problem 458
Let $G$ be a finite group of order $217$.
(a) Prove that $G$ is a cyclic group.
(b) Determine the number of generators of the group $G$.
Add to solve later Surjective Group Homomorphism to $\Z$ and Direct Product of Abelian Groups
Problem 342
Let $G$ be an abelian group and let $f: G\to \Z$ be a surjective group homomorphism.
Prove that we have an isomorphism of groups:
\[G \cong \ker(f)\times \Z.\]
Add to solve later Pullback Group of Two Group Homomorphisms into a Group
Problem 244
Let $G_1, G_1$, and $H$ be groups. Let $f_1: G_1 \to H$ and $f_2: G_2 \to H$ be group homomorphisms.
Define the subset $M$ of $G_1 \times G_2$ to be
\[M=\{(a_1, a_2) \in G_1\times G_2 \mid f_1(a_1)=f_2(a_2)\}.\]
Prove that $M$ is a subgroup of $G_1 \times G_2$.
Add to solve later 
