Inclusion Diagram for Cyclic, Abelian, Nilpotent, and Solvable Groups

Recall the following definitions:

• 1) A group $G$ is said to be Cyclic if it can be generated by a single element. That is, there exists an element $a \in G$ such that $G = (a)$ where $(a) = \{ a^n : n \in \mathbb{Z} \}$.

• 2) A group $G$ is said to be Abelian if for all $a, b \in G$ we have that $ab = ba$, that is, every element in $G$ commutes with every element in $G$.

• 3) A group $G$ is said to be Nilpotent of Class $c$ if it has a terminating upper ascending central series of length $c$, $\{ 1 \} = Z_0 \triangleleft Z_1 \triangleleft ... \triangleleft Z_c = G$ where $Z_1 = Z(G)$ and $Z_{i+1}$ is defined such that $Z_{i+1}/Z_i = Z(G/Z_i)$.

• 4) A group $G$ is said to be Solvable if $G$ has a subnormal series $\{ 1 \} = G_0 \trianglelefteq G_1 \trianglelefteq ... \trianglelefteq G_k = G$ whose factors $G_{i+1}/G_i$ are all abelian groups.

Below is a chart showing the inclusions of the sets of cyclic, abelian, nilpotent, and solvable groups.