Index of Common Groups

# Index of Common Groups

Group Name | Group Symbol | Group Operation | Description | Size of Group | Abelian/Non-Abelian |
---|---|---|---|---|---|

Trivial Group | $(\{ e \}, \cdot)$ | - | The simplest group consisting of only one element. | $1$ | Abelian |

The Additive Group of $\mathbb{C}$ | ($\mathbb{C}, +)$ | Addition | The group of complex numbers under addition. | (Uncountably) Infinite | Abelian |

The Additive Group of $\mathbb{R}$ | ($\mathbb{R}, +)$ | Addition | The group of real numbers under addition. | (Uncountably) Infinite | Abelian |

The Additive Group of $\mathbb{Q}$ | ($\mathbb{Q}, +)$ | Addition | The group of rational numbers under addition. | (Countably) Infinite | Abelian |

The Additive Group of $\mathbb{Z}$ | ($\mathbb{Z}, +)$ | Addition | The group of integers under addition. | (Countably) Infinite | Abelian |

The Additive Group of Multiples of $n$ | $(n\mathbb{Z}, +)$ | Addition | The group of all multiples of $n$ under addition. | (Countably) Infinite | Abelian |

The Additive Group of Integers Modulo $n$ | $(\mathbb{Z}/n\mathbb{Z}, +)$ | Addition Modulo $n$ | The group obtained by splitting $\mathbb{Z}$ into equivalence classes where $x \equiv y$ if and only if $n \mid (x - y)$. | $n$ | Abelian |

The Additive Group of a Vector Space $V$ | $(V, +)$ | Vector Addition | The underlying additive group of the vector space $V$. | - | Abelian |

The Multiplicative Group of $\mathbb{C} \setminus \{ 0 \}$ | ($\mathbb{C}^{\times}, \cdot)$ | Multiplication | The group of nonzero complex numbers under multiplication. | (Uncountably) Infinite | Abelian |

The Multiplicative Group of $\mathbb{R} \setminus \{ 0 \}$ | ($\mathbb{R}^{\times}, \cdot)$ | Multiplication | The group of nonzero real numbers under multiplication. | (Uncountably) Infinite | Abelian |

The Multiplicative Group of $\mathbb{Q} \setminus \{ 0 \}$ | ($\mathbb{Q}^{\times}, \cdot)$ | Multiplication | The group of nonzero rational numbers under multiplication. | (Countably) Infinite | Abelian |

The Multiplicative Group of Invertible Integers Modulo $n$ | ($(\mathbb{Z}/n\mathbb{Z})^{\times}, \cdot)$ | Multiplication Modulo $n$ | The group of integers in $\{ 1, 2, ..., n \}$ that has multiplicative inverses modulo $n$. | $\phi(n)$ | Abelian |

Continuous Real-Valued Functions on $[a, b]$ | $(C[a, b], +)$ | (Pointwise) Function Addition | The group of continuous functions on the closed bounded interval $[a, b]$. | Infinite | Abelian |

$n$-Times Differentiable Real-Valued Functions on $[a, b]$ | The group of $n$-times differentiable functions on the closed bounded interval $[a, b]$. | Function Addition | $(C^n[a, b], +)$ | Infinite | Abelian |

The Additive Group of $m \times n$ Matrices | $(M_{mn}, +)$ | Matrix Addition | The group of all $m \times n$ matrices with entries sin $\mathbb{R}$. | Infinite | Abelian |

The General Linear Group of Order $n$ | $(\mathrm{GL}_n(\mathbb{R}), \cdot)$ | Matrix Multiplication | The group of all $n \times n$ invertible matrices with entries in $\mathbb{R}$. | Infinite | Abelian if $n = 1$, Non-Abelian if $n \geq 2$ |

The Special Linear Group of Order $n$ | $(\mathrm{SL}_n(\mathbb{R}), \cdot)$ | Matrix Multiplication | The group of all $n \times n$ invertible matrices with entries in $\mathbb{R}$ and with determinant $1$. | Infinite | Abelian if $n = 1$, Non-Abelian if $n \geq 2$ |

Cyclic Groups | $G = \langle a \rangle$ | - | Any group that is generated by a single element $a \in G$, that is, every $g \in G$ is of the form $g = a^n$ for some $n \in \mathbb{Z}$. | - | Abelian |

Cyclic Group of Order $n$ | $Z_n$ | - | The group of size $n$ generated by a single element $a \in G$ | $n$ | Abelian |

Cyclic Group of Order $p$ (prime) | $Z_p$ | - | The group of size $p$ (prime) generated by a single element. Any group of order $p$ is necessarily isomorphic to $Z_p$. | $p$ | Abelian |

Direct Product Group of the Groups $(G, \circ)$ and $(H, *)$ | $(G \times H, -)$ | Componentwise Application of Corresponding Group Operations | The group of the cartesian product $G \times H = \{ (g, h) : g \in G, h \in H \}$ with the operation defined by $(g_1, h_1)(g_2, h_2) = (g_1 \cdot g_2, h_1 * h_2)$. | $\mid G \mid \mid H \mid$ | Abelian if and only if both $G$ and $H$ are abelian. |

The Klein Four-Group | $K_4$ | - | The group isomorphic to $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ | $4$ | Abelian |

The Center of a Group $(G, \cdot)$ | $(Z(G), \cdot)$ | - | The group of all elements in $G$ that commute with every other element in $G$. | - | Abelian |

Symmetric Group on $n$-Elements | $(S_n, \circ)$ | Composition (of permutations) | The group of all permutations of the set $\{ 1, 2, ..., n \}$. | $n!$ | Abelian if $n = 1, 2$, Non-Abelian if $n \geq 3$. |

Permutation Group on a Set $X$ | $(S_X, \circ)$ | Composition (of permutations) | The group of all permutations of the set $X$. | - | Abelian if $|X| \leq 2$, Non-Abelian if $|X| \geq 3$ or if $X$ is an infinite set. |

Dihedral Group of the Regular $n$-gon ($n \geq 3$) | $(D_n, \circ)$ | Composition (of symmetries) | The group of all rigid translations of the regular $n$-gon. | $2n$ | Non-Abelian |

The Alternating Group on $n$-Elements | $(A_n, \circ)$ | Composition (of even permutations) | The group of all even permutations of the set $\{ 1, 2, ..., n \}$. | $\displaystyle{\frac{n!}{2}}$ | Abelian if $n = 1, 2, 3$, Non-Abelian if $n \geq 4$. |

Automorphism Group of a Group $G$ | $(\mathrm{Aut}(G), \circ)$ | Composition (of Automorphisms) | The group of all automorphisms (i.e., isomorphisms from $G$ to $G$). | - | Abelian |

The Free Group on $n$ Generators ($n \geq 1$) | $F_n$ | (Reduced) Concatenation of Words | The group of all finite words constructed by the $n$ elements $x_1, x_2, ..., x_n$ and $x_1^{-1}, x_2^{-1}, ..., x_n^{-1}$, with the identity element being the empty word. | Infinite | Abelian if $n = 1$, Non-Abelian if $n \geq 2$. |

The Free Group on a Nonempty Set $X$ | $F(X)$ | (Reduced) Concatenation of Words | The group of all finite words constructed by the sets of symbols $X$ and $X^{-1}$ (where $X^{-1}$ is disjoint from $X$ and in bijection with $X$), with the identity element being the empty word. | Infinite | Abelian if $|X| = 1$, Non-Abelian if $|X| \geq 2$. |