Two commutative groups $G$ and $G^{\prime}$ are said to be isomorphic provided that there exists a bijective correspondence between their elements, $f: G \rightarrow G^{\prime}$, which preserves the group operations [i.e., $f(x+y)=f(x)+$ $f(y), f(0)=0$, and $f(-x)=-f(x)]$. Such groups can be considered as "essentially the same": the name of an element $x$ of $G$ is just changed to $f(x)$.
(1) Prove that there exists essentially just one commutative group of 1 element (i.e., any two one-element groups are isomorphic).
(2) Prove that there exists essentially just one commutative group of 2 elements [i.e., any two-element group is isomorphic to $\left(Z_2,+\right)$ ].
(3) Let $K$ be a subgroup of a commutative group $G$. Prove that there is essentially just one group $G / K$ of coset leaders (i.e., various choices of coset leaders lead to isomorphic groups).
Describe the group $\mathbf{Z}_4 / K$, where $K=\{0,2\}$.
(4) Prove that the group $(\exp M, \Delta)$ of $6 \mathrm{E}$ is isomorphic to $\left(\mathbb{Z}_2^n,+\right)$.