Uncombinatorial Abstract Algebra

semigroup

SG := {G = ⟨ \hat{G}, \cdot ⟩ | \cdot \in {\hat{G}}^{\hat{G} \times \hat{G}} \land (\forall a, b, c \in \hat{G}) ((a \cdot b) \cdot c) = (a \cdot (b \cdot c))}

$a \cdot b ⇝ a b$

$g \in \hat{G} ⇝ g \in G$

monoid

MN := {G \in SG | (\exists e \in G) (\forall g \in G) (e g = g e = g)}

$⊢ w . d . (E : G \mapsto e)$

$◂ (\exists e, e^{'} \in G) (\forall g \in G) (e g = g e = g \land e^{'} g = g e^{'} = g) \Rightarrow e = e e^{'} = e^{'} ▸$

$E (G) ⇝ e$

group

G := {G \in MN | (\forall g \in G) (\exists g^{'} \in G) (g^{'} g = g g^{'} = e)}

$⊢ w . d . (INV : g \mapsto g^{'})$

$◂ (\exists g_{1}, g_{2} \in G) (g_{1} g = g g_{1} = e \land g_{2} g = g g_{2} = e) \Rightarrow g_{1} = g_{1} (g g_{2}) = (g_{1} g) g_{2} = g_{2} ▸$

$INV (g) ⇝ g^{- 1}$

$⊢ (g^{- 1})^{- 1} = g$

$⊢ (a b)^{- 1} = b^{- 1} a^{- 1}$

$⊢ w . d . (a^{k} | k \in Z)$

$⊢ G \in SG \land (\exists e \in G) (\forall g \in G) (e g = g) \land (\forall g \in G) (\exists g^{'} \in G) (g^{'} g = e) \to G \in G$

$◂ g g^{'} = e g g^{'} = (g^{″} g^{'}) g g^{'} = g^{″} (g^{'} g) g^{'} = g^{″} e g^{'} = g^{″} g = e, g e = g (g^{'} g) = (g g^{'}) g = e g = g ▸$

subgroup

$G \in G \land H \in G \land \hat{H} \subseteq \hat{G} ⇝ H \leq G$

$⊢ G \in G \land \hat{H} \subseteq \hat{G} \land (\forall a, b \in \hat{H}) (a b \in \hat{H} \land a^{- 1} \in \hat{H}) \to H \in G$

$⊢ G \in G \land \hat{H} \subseteq \hat{G} \land (\forall a, b \in \hat{H}) (a b^{- 1} \in \hat{H}) \to H \in G$

$⊢ G \in G \land \hat{H} \subseteq \hat{G} \land | \hat{H} | < \infty \land (\forall a, b \in \hat{H}) (a b \in \hat{H}) \to H \in G$

center

$C : G \mapsto {c \in G | (\forall g \in G) (c g = g c)}$

$⊢ C (G) \leq G$

$⊢ \emptyset ⊊ H \subseteq {H | H \leq G} \to ⋂ H \leq G$

$⊢ w . d . (⟨ S ⟩ := H | H \leq G \land S \subseteq \hat{H} \land (\forall H^{'} \leq G) (S \subseteq \hat{H^{'}} \to H \leq H^{'}))$

$⊢ \hat{⟨ S ⟩} = {\prod_{i = 1}^{n} a_{i}^{ε_{i}} | n \in N^{*} \land a_{i} \in S \land ε_{i} \in {\pm 1}}$

$⟨ {a} ⟩ ⇝ ⟨ a ⟩$

period

$o (a) := inf {n \in N^{*} | a^{n} = e}$

coset

${a h | h \in H} ⇝ a H$

$⊢ H \leq G \land g \in G \to (g \in H \leftrightarrow g H = H)$

$⊢ | a H | = | H |$

$⊢ a H \neq b H \to a H \cap b H = \emptyset$

$◂ a H \neq b H \land a h_{1} = b h_{2} \in a H \cap b H \Rightarrow b^{- 1} a H \neq H \land b^{- 1} a = h_{2} h_{1}^{- 1} \in H \Rightarrow b^{- 1} a \notin H \land b^{- 1} a \in H \Rightarrow! ▸$

$G /_{L} H := {a H | a \in G}$

$[G : H] := | G /_{L} H |$

Lagrange theorem

⊢ H \leq G \to | G | = | H | [G : H]