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 $|X| \geq 2$.