Group Theory/Groups, subgroups and constructions

Henceforth, we shall sometimes refer to the group operation of a group simply as the "operation".

Note: Often, the explicit notation for the group operation is omitted and the product of two elements is denoted solely by juxtaposition.

Subgroups with the inclusion map $\iota :H\to G$ represent subobjects of a group.

Exercises

1. Make explicit the proof of right-cancellation ("right-cancellation" means $y*x=z*x\Rightarrow y=z$).
2. Let $G$ be a group, and let $H,J\le G$ be subgroups such that neither $H\subseteq J$ nor $J\subseteq H$. Prove that $H\cup J$ is not a subgroup of $G$.
3. Let $G=\{1,-1\}$ together with the operation $1*1=1=-1*-1$, $1*-1=-1*1=-1$.
   1. Prove in detail that $G$, together with the operation $*$, is a group.
   2. Prove that in $G\times G$, there exists a subgroup which is not equal to $H\times L$ with subgroups $H,L\le G$.