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 $$1 * 1 = 1 = -1 * -1$$, $$1 * -1 = -1 * 1 = -1$$.
 * 4) Prove in detail that $$G$$, together with the operation $$*$$, is a group.
 * 5) Prove that in $$G \times G$$, there exists a subgroup which is not equal to $$H \times L$$ with subgroups $$H, L \le G$$.