Question about Abstract Algebra Notation for Abelian Group I've never used this forum before but I had a question about the formatting of one my homework problems.
Prove: If $H$ is an Abelian subgroup of a group $G$, then $\langle H, Z(G)\rangle$ is an Abelian group.
I suppose my confusion is with the use of generator notation containing two arguments.
I understand that the problem is essentially asking me to show if we have an Abelian subgroup $H$ and the center of $G,$ some combination of their elements must be Abelian.
In some kind of psuedo-proof logic:

*

*We know for all $x,y$ in $H$, $xy=yx$

*We know for all $x$ in $Z(G)$ and all $y$ in $G$, $xy=yx$
Is $\langle H, Z(G)\rangle$ a generator for a group with elements that are pairs $(x,y)$?
Thank you in advance if you can help me in any way. I looked through D&F and online but wasn't able to find much or wasn't equipped mentally to know what to look for.

Thank you all so much.
Here was my first stab at this proof.
Let G be a group.
Let H be an Abelian subgroup of G.
By the definition of the center of a group, Z(G) = {x∈G|∀y∈G (xy=yx)}
The generator <H, Z(G)> = <H ∪ Z(G)> = H ∪ Z ∪ {hz | h∈H, z∈Z(G)}
Let x,y ∈ H. Because H ≤ G, x,y ∈ G.
Any element z∈Z(G) commutes with all elements of G => zx = xz and zy=yz
Because all elements in the set generated by <H, Z(G)> commute, then <H, Z(G)> is Abelian.
