Abstract Algebra Dictionary I

Definitions

• A function $f: S \rightarrow T$ is:

  1. Injective if $\forall b \in T$, there is at most one $a \in S$ s.t. $f(a)=b$.

  2. Surjective if $\forall b \in T$, there is at least one $a \in S$ s.t. $f(a)=b$.

  3. Bijective if $\forall b \in T$, there is exactly one $a \in S$ s.t. $f(a)=b$.

• For functions $f_1: S \rightarrow T$ and $f_2: T \rightarrow R$, the composition $f_2 \circ f_1: S \rightarrow R$ is defined by $\forall a \in S: (f_2 \circ f_1)(a) = f_2(f_1(a))$.

• A function $f: S \rightarrow T$ is invertible if $\exists \, g: T \rightarrow S$ s.t. $\text{id}_S = g \circ f : S \rightarrow S$ and $\text{id}_T = f \circ g : T \rightarrow T$.

• Two subsets $S_1$ and $S_2$ are disjoint if $S_1 \cap S_2 = \varnothing$.

• A set $S$ is the disjoint union of subsets $S_1, S_2, \dots, S_n \subset S$ if:

  1. $S = S_1 \cup S_2 \cup \dots \cup S_n$

  2. $\forall i \neq j, S_i \cap S_j = \varnothing$ (all subsets are pairwise disjoint).

• An equivalence relation on a set $S$ is a rule $\sim$ satisfying:

  1. Reflexivity: $\forall a \in S, a \sim a$

  2. Symmetry: $\forall a, b \in S$, if $a \sim b$ then $b \sim a$

  3. Transitivity: $\forall a, b, c \in S$, if $a \sim b$ and $b \sim c$, then $a \sim c$.

• Given an equivalence relation $\sim$ on $S$ and an element $a \in S$, the equivalence class of $a$ is $[a] = S_a := \{b \in S : b \sim a\}$.

• Given an equivalence relation $\sim$ on $S$, the set of distinct equivalence classes (quotient set) is $S/\sim \;:= \{[a_1], [a_2], \dots, [a_n]\} = \{S_{a_1}, S_{a_2}, \dots, S_{a_n}\}$.

• Given $a, b \in \mathbb{Z}$ with $a \neq 0$, $a$ divides $b$ (denoted $a \mid b$) if $\exists \, c \in \mathbb{Z}$ s.t. $ac = b$. Here, $a$ is called a divisor of $b$.

• For $m \in \mathbb{Z}$, the relation congruence modulo $m$ ($a \sim b$) is defined by $m \mid (a - b)$.

• The integers modulo $m$ is defined as $\mathbb{Z}/m\mathbb{Z} := \{[0], [1], \dots, [m-1]\}$.

• A permutation of a set $S$ is a bijective function from $S$ to itself.

• A group is a set $G$ together with a composition law $G \times G \rightarrow G, (g_1, g_2) \mapsto g_1 g_2$, satisfying:

  1. Identity: $\exists \, e \in G$ s.t. $\forall g \in G, eg = ge = g$

  2. Inverse: $\forall g \in G, \exists \, h \in G$ s.t. $gh = hg = e$

  3. Associativity: $\forall g_1, g_2, g_3 \in G, g_1(g_2 g_3) = (g_1 g_2)g_3$.

  If in addition $\forall g_1, g_2 \in G, g_1 g_2 = g_2 g_1$ (commutativity), then $G$ is an abelian group.

• The order of a group $G$ is the number of elements in the underlying set, $\#G$.

• The order of an element $g \in G$ is the smallest positive integer $n \in \mathbb{N}$ s.t. $g^n = e$.

• A group $G$ is cyclic if $\exists \, g \in G$ s.t. $G = \{g^k : k \in \mathbb{Z}\}$. The element $g$ is called a generator of $G$.

• The cyclic group of order $n$ is $C_n := \{e = g^0, g^1, g^2, \dots, g^{n-1}\}$.

• Let $G$ and $G'$ be groups. A homomorphism is a function $\phi: G \rightarrow G'$ s.t. $\forall g_1, g_2 \in G, \phi(g_1 g_2) = \phi(g_1)\phi(g_2)$.

• An isomorphism is a bijective group homomorphism.

• Let $G$ be a group. A subgroup of $G$ is a subset $H \subset G$ s.t. $H$ is itself a group under the same composition law as $G$.

• Given $g \in G$, the cyclic subgroup generated by $g$ is $\langle g \rangle := \{e = g^0, g^1, \dots, g^{n-1}\}$.

• Let $\phi: G \rightarrow G'$ be a group homomorphism. The kernel of $\phi$ is $\ker(\phi) := \{g \in G : \phi(g) = e'\}$, which is a subgroup of $G$.

• Let $G$ be a group and $H \subset G$ a subgroup. For each $g \in G$, the (left) coset of $H$ attached to $g$ is $gH := \{gh : h \in H\} \subset G$.

• Let $G$ be a group and $H \subset G$ a subgroup. The index of $H$ in $G$, denoted $(G : H)$, is the number of distinct cosets of $H$ in $G$.

Propositions and Theorems

• A function $f: S \rightarrow T$ is invertible if and only if it is bijective.

• Let $\sim$ be an equivalence relation on a set $S$. Then for all $a, b \in S$, either $[a] = [b]$ (i.e., $S_a = S_b$) or $[a] \cap [b] = \varnothing$ (i.e., $S_a \cap S_b = \varnothing$).

• For an equivalence relation $\sim$ on a set $S$:

  1. The distinct equivalence classes give a disjoint union: $S = S_{a_1} \cup S_{a_2} \cup \dots \cup S_{a_n}$.

  2. If $c \in S_a$, then $S_a = S_c$.

• For any $m \in \mathbb{Z}$, the set of integers is the disjoint union of residue classes modulo $m$: $\mathbb{Z} = [0] \cup [1] \cup \dots \cup [m-1]$.

• Let $G$ be a group:

  1. $G$ has a unique identity element $e$.

  2. Each element $g \in G$ has a unique inverse $g^{-1}$, and $(gh)^{-1} = h^{-1}g^{-1}$.

  3. The inverse of the inverse is the original element: $(g^{-1})^{-1} = g$.

  4. Cancellation law holds: if $gh = gk$, then $h = k$.

• Let $G$ be a group and $g \in G$. If $g^n = e$, then the order of $g$ divides $n$ ($\text{ord}(g) \mid n$).

• A finite group $G$ is cyclic if and only if there exists $g \in G$ such that $\text{ord}(g) = \#G$.

• Let $\phi: G \rightarrow G'$ be a group homomorphism:

  1. $\phi(e_G) = e_{G'}$.

  2. $\forall g \in G, \phi(g^{-1}) = \phi(g)^{-1}$.

• Let $\phi: G \rightarrow G'$ be a group homomorphism. The kernel of $\phi$, defined as $\ker(\phi) := \{g \in G : \phi(g) = e'\}$, is a subgroup of $G$.

• Let $G$ be a group and $H \subset G$ a subgroup. A coset $gH$ is a subgroup of $G$ if and only if $g \in H$ (in which case $gH = H$).

• Let $G$ be a finite group and $H \subset G$ a subgroup:

  1. Every element of $G$ is contained in some coset of $H$ ($\forall g \in G, g \in gH$).

  2. For all $g \in G$, $\#(gH) = \#H$.

  3. For all $g_1, g_2 \in G$, either $g_1 H = g_2 H$ or $g_1 H \cap g_2 H = \varnothing$.

• (Lagrange's Theorem) Let $G$ be a finite group and $H \subset G$ a subgroup. Then: $\#G = \#H \cdot (G : H)$

• For a finite group $G$ and subgroup $H \subset G$:

  1. If $G$ is a finite group and $H \subset G$ is a subgroup, then $\#H \mid \#G$.

  2. If $G$ is a finite group and $g \in G$, then $\text{ord}(g) \mid \#G$.

  3. If $\#G = p$ is a prime number, then $G$ is cyclic (and $G \simeq C_p$).

• Let $\phi: G \rightarrow G'$ be a group homomorphism. For any $g' \in \text{im}(\phi)$, the preimage $\phi^{-1}(g') := \{h \in G : \phi(h) = g'\}$ is a left coset of $\ker(\phi)$. Specifically, if $\phi(h_1) = g'$, then $\phi^{-1}(g') = h_1 \ker(\phi)$.