Intereting Posts

Proof that if group $G/Z(G)$ is cyclic, then $G$ is commutative
Suppose that R is a commutative ring with unity such that for each $a$ in $R$ there is an integer $n > 1\mid a^n =a$. Every prime ideal is maximal?
Triangles area question
What does “sets of arbitrarily large measure” mean — question about $L_p$ embeddings
Why can't I combine complex powers
Conditional mean on uncorrelated stochastic variable 2
Are continuous functions with compact support bounded?
What are the surfaces of constant Gaussian curvature $K > 0$?
A converse proposition to the Mean Value Theorem
Infinite sum of reciprocals of pentagonal numbers
Real Analysis Convergence question
Subgroups of Symmetric groups isomorphic to dihedral group
Does a convergent power series on a closed disk always converge uniformly?
Determining possible minimal polynomials for a rank one linear operator
If $f(x)$ is continuous on $$ and $M=\max \; |f(x)|$, is $M=\lim \limits_{n\to\infty} \left(\int_a^b|f(x)|^n\,\mathrm dx\right)^{1/n}$?

I am not sure how to show the second part. Isn’t it obvious from the fact that G and H are infinite?

- Any finitely generated subgroup of $(\mathbb{Q},+)$ is cyclic.
- $G$ be a free abelian group of rank $k$ , let $S$ be a subset of $G$ that generates $G$ then is it true that $|S| \ge k$?
- If $X$ is the set of all group elements of order $p$, and $X$ is finite, then $\langle X \rangle$ is finite
- Can I recover a group by its homomorphisms?
- Must $k$-subalgebra of $k$ be finitely generated?
- $\mathbb Z$ is not finitely generated?

- Any finitely generated subgroup of $(\mathbb{Q},+)$ is cyclic.
- How can I find an element $x\not\in\mathfrak mM_{\mathfrak m}$ for every maximal ideal $\mathfrak m$
- What are the symmetries of a colored rubiks cube?
- the automorphism group of a finitely generated group
- An ideal which is not finitely generated
- free subgroups of $SL(2,\mathbb{R})$
- Isometries of a hyperbolic quadratic form
- If $X$ is the set of all group elements of order $p$, and $X$ is finite, then $\langle X \rangle$ is finite
- Is the Cayley graph of a word-hyperbolic group a CAT(0) metric space?
- Finitely generated graded modules over $K$

The following was a bit too long for a comment so I wrote it as an answer:

In my opinion it is better to look at this problem in a pure structure theoretical fashion, since your assertion is true for any pair of connected infinite locally finite graphs. So nothing special about the graphs being Cayley graphs of some groups.

Hence: Let $X$ and $Y$ be two connected infinite locally finite graphs. Then their product $X \times Y$ has exactly one end.

There are lots of “different products” one could form with two given graphs. In this case we have $V(X \times Y) = V(X) \times V(Y)$ and the adjacency is defined as follows:

$(x_1, y_1)$ is adjacent to $(x_2, y_2)$ iff $x_1 = x_2$ and $y_1$ is adjacent to $y_2$ or vice versa.

Let now $B$ be some ball of finite radius in $X \times Y$ and let $z,w$ be two vertices outside of $B$ and each inside some *infinite* component. If you find a path connecting them outside of $B$ you are done.

First think about the example $\mathbb{Z} \times \mathbb{Z}$ and some ball in its Cayley graph (w.r.t some generating set). Why can you connect any two vertices outside this ball without touching the ball (consider the definition of the adjacency in such product graphs at the same time)?

I am confident that you can write it down in general.

- Basis for $\mathbb{R}$ over $\mathbb{Q}$
- Intuition behind Conditional Expectation
- Are $X$ and $X+Y$ independent, if $X$ and $Y$ are independent?
- Numerical solution to a system of second order differential equations
- Continuous image of a locally connected space which is not locally connected
- Offseting a Bezier curve
- Computing geodesics (or shortest paths)
- Why is “the set of all sets” a paradox?
- Compact space, continuous dynamical system, stationary point
- What is the Riemann-Zeta function?
- Proof of an inequality in a triangle
- How to show that the geodesics of a metric are the solutions to a second-order differential equation?
- Show that $A=\{ \frac{m}{2^n}:m\in \mathbb {Z},n\in \mathbb {N} \} $ is dense in $\mathbb {R}$
- Can $A, B$ fail to commute if $e^A=e^B=e^{A+B}=id$?
- What consistent rules can we use to compute sums like 1 + 2 + 3 + …?