# 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 $G$.

## 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:

\[G \cong \ker(f)\times \Z.\]

## 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 $G_1 \times G_2$.

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