(** The following is a commented reading of the formalization of finite groups provided by the MathComp library. The goal is to explain the definitions and the notations, and to replay some of the proofs in a less compact way for didactic purposes. I used the following references for the mathematics %\cite{kurzweil2004,oggier2011}% and received comments from Cyril Cohen. *) (** * Basics About Groups *) (** The formalization of finite groups is essentially built on top of %\texttt{finset.v}%, the formalization of finite sets, hence the following sequence of %\lstinline!Import!%s. *) From Ssreflect Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice fintype. From MathComp Require Import div path bigop prime finset fingroup. (** In %\mathcomp%, finite groups are subgroups'' of a container group''. The first part of the container group structure can be found in the %\lstinline!Record mixin_of!% (file: %{\tt fingroup.v}%; %\lstinline!Module FinGroup!%): a binary operation, a special element (the neutral), a unary operation (inverse), associativity, left identity, involution, antimorphism (i.e., $(xy)^{-1} = y^{-1}x^{-1}$). Such a structure is not a group because $x^{-1} x = 1$ may not hold. *) Print FinGroup.mixin_of. (** The carrier of a container group is put together with (1) the assumption that it satisfies the structure above and (2) the assumption that the carrier is a finite type. This forms a %\lstinline!baseFinGroupType!%. *) Print FinGroup.base_type. Print baseFinGroupType. (** Then, the %\lstinline!baseFinGroupType!% is put together with the following law: $\forall x, x^{-1} x = 1$. This gives a group that will act as a container group'' hereafter. *) Print FinGroup.type. Print finGroupType. (** Finally, a finite group is defined as a set of elements from a finite container group together with the assumption that it is a %\lstinline!group_set!%, i.e., that it contains the neutral element and that it is stable by the binary operation. It is worth noticing that the definitive definition of a finite group appears very late in the %{\tt fingroup.v}% file (almost in the middle of a 3,000 lines file). *) Print group_set. Print group_type. (** How to declare a finite group? First declare a finite container group %\lstinline!gT : finGroupType!% and then a finite group using the dedicated notation %\lstinline!G : {group gT}!% (scope: %\lstinline!type_scope!%, notation for %\lstinline!group_of (Phant gT)!%). %\lstinline!{group _}!% has type %\lstinline!predArgType!% which means that it comes with the generic notation %\lstinline!\in!%. *) Section group_example. Variable gT : finGroupType. Variable G : {group gT}. (** Groups enjoy the following notations. *) Local Open Scope group_scope. Check (1 : gT). Fail Check (1 : G). Check (1 \in G). Check (1 * 1 : gT). Check (1 ^-1 : gT). Lemma neutral_in_group : 1 \in G. Proof. Check group1. rewrite group1. done. Qed. Lemma neutral_neutral_in_group : 1 * 1 \in G. Proof. Search _ left_id mulg in fingroup. rewrite mul1g. rewrite group1. done. Qed. (** Point multiplication and inverse are lifted to set of points. *) Check set_mulg. Check set_invg. (** For two nonempty subsets $A,B$ of $G$, let $AB := \{ ab | a \in A, b \in B\}$. $AB$ is the %\mydef{(complex) product}% of $A$ and $B$. When $A = \{ a \}$, we write $aB$ instead of $AB$. A product is not necessarily a group (the multiplication needs to commute for that) but it is at least a %\lstinline!group_set_of_baseGroupType!%. *) Fail Check (G * G : {group gT}). Check (G * G : group_set_of_baseGroupType gT). Set Printing All. Check (1 = [set 1] :> {set gT}). Unset Printing All. Lemma neutral_in_group_group : 1 \in G * G. Proof. Check mulSGid. rewrite mulSGid. rewrite group1. done. rewrite subxx. done. Qed. (** Let $A$ and $B$ be subgroups of $G$. Then $AB$ is a subgroup of $G$ iff $AB = BA$. *) Lemma group_set_group_group : group_set (G * G). Proof. Search _ group_set reflect in fingroup. apply/comm_group_setP. Search (commute _ _). apply commute_refl. Qed. (** Let $U$ be a subgroup of $G$ and $x \in G$. The product $U x$ is a %\mydef{right coset}% of $U$ in $G$. The right coset of $H$ by $x$ is noted %\lstinline!H :* x!% in %\mathcomp\% (notation scope: %\lstinline!group_scope!%; file: %{\tt fingroup.v}%). There is another definition of right cosets (%\lstinline!Definition rcoset!%) that is proved equivalent (%\lstinline!Lemma rcosetE!%). *) Variable x : gT. Variable H : {group gT}. Check (H :* x : {set gT}). Locate ":*". (** The map $H \to Ha; h \mapsto ha$ is injective. Thus a coset $Ha$ has cardinal $|H|$. *) Lemma mycard_rcoset : #|H :* x| = #| H |. Proof. Check card_rcoset. Check card_imset. rewrite -[in X in _ = X](@card_imset _ _ (mulg^~ x)); last first. Search _ left_injective mulg. exact: mulIg. rewrite -rcosetE. rewrite /rcoset. done. Qed. (** The set of the right cosets of $H$ by elements of $G$ is denoted by %\lstinline!rcosets H G!% (file: %{\tt fingroup.v}%). *) Check (rcosets H G : {set {set gT}}). Check (rcosets H G). (** If the set of right cosets of $U$ in $G$ is finite then the number of right cosets of $U$ in $G$ is the %\mydef{index}% of $U$ in $G$, denoted by %$|G : U|$% (%\lstinline!Definition indexg!%; file %{\tt fingroup.v}%). *) Print indexg. End group_example. (** * Lagrange's Theorem *) (** Lagrange's theorem is already proved in %{\tt fingroup.v}%. In the following, we replay this proof in a less compact way. *) Check LagrangeI. Section myLagrange. Variable gT : finGroupType. Local Open Scope group_scope. Variable (H G : {group gT}). Hypothesis HG : H \subset G. (** The relation $xy^{-1} \in H$ is an equivalence relation. The equivalence class of $x$ (the set of $y$ such that $xy^{-1} \in H$) is actually the right coset $Hx$. The set of cosets forms a partition of $G$. We first prove this fact. *) Print equivalence_partition. Lemma rcosets_equivalence_partition : rcosets H G = equivalence_partition [rel x y | x * y^-1 \in H] G. Proof. apply/setP => /= X. case/boolP : (X \in equivalence_partition _ _). case/imsetP => x Hx. move=> ->. apply/rcosetsP. exists x => //. apply/setP => y. rewrite inE /=. case/boolP : (_ \in _ :* _). case/rcosetP => z Hz. move=> ->. rewrite invMg. rewrite mulgA. rewrite mulgV. rewrite mul1g. rewrite groupVr //. rewrite andbT. rewrite groupM //. move/subsetP : HG. by apply. apply: contraNF. case/andP => Hy xy. apply/rcosetP. exists (y * x^-1). rewrite groupVl //. rewrite invMg. by rewrite invgK. rewrite -mulgA. rewrite mulVg. by rewrite mulg1. apply: contraNF. case/rcosetsP => x Hx ->. apply/imsetP. exists x => //. apply/setP => /= y. rewrite inE. case/boolP : (_ && _). case/andP => Hy xy. apply/rcosetP. exists (y * x^-1). rewrite groupVl //. rewrite invMg. by rewrite invgK. rewrite -mulgA. rewrite mulVg. by rewrite mulg1. apply: contraNF. case/rcosetP => z Hz ->. apply/andP; split. rewrite groupM //. move/subsetP : HG. by apply. rewrite invMg. rewrite mulgA. rewrite mulgV. rewrite mul1g. by rewrite groupVr. Qed. Lemma myrcosets_partition : partition (rcosets H G) G. Proof. rewrite rcosets_equivalence_partition. apply/equivalence_partitionP. move=> x y z Hx Hy Hz /=. split. rewrite mulgV. by rewrite group1. move=> xy. case/boolP : (y * _ \in _) => yz. move: (groupM xy yz). rewrite mulgA. rewrite -(mulgA x). rewrite mulVg. by rewrite mulg1. apply: contraNF yz => yz. move/groupVr in xy. move: (groupM xy yz). rewrite invMg. rewrite invgK. rewrite mulgA. rewrite -(mulgA y). rewrite mulVg. by rewrite mulg1. Qed. (** Lagrange's theorem follows from the fact that the right cosets form a partition of $G$ and that each coset has the same cardinal as $H$. *) Lemma myLagrange : #| G | = (#|H| * #|G : H|)%(*coq_*)nat. Proof. have -> : #|G| = \sum_(U in rcosets H G) #|U|. move: myrcosets_partition. move/card_partition. done. transitivity (\sum_(U in rcosets H G) #|H|). apply eq_bigr => /= U HU. case/rcosetsP : HU => u hG ->. by rewrite mycard_rcoset. rewrite big_const. rewrite iter_addn. rewrite addn0. rewrite -/(#|G : H|). done. Qed. End myLagrange. (** * Normal Subgroups *) (** For $x, a \in G$ set $x^a := a^{-1}xa$. This element $x^a$ is the %\mydef{conjugate}% of $x$ by $a$. (notation: %\lstinline!x ^ y!%; notation scope: %\lstinline!group_scope!%; file: %{\tt fingroup.v}%). *) Print conjg. Locate "^". (** Sample property: $x^1 = x$. *) Check conjg1. (** For $g \in G$ we set $B^g := g^{-1} B g$ and say that $B^g$ is the conjugate of $B$ by $g$. In %\ssreflect%, the conjugate of $H$ by $x$ is denoted by %\lstinline!H :^ x!%. *) Print conjugate. (** The %\mydef{normalizer}% of $A$? $\{x | A^x \subseteq A\}$ (Notation: %\lstinline!'N(A)!%; definition; file: %\texttt{fingroup.v}%). *) Print normaliser. Locate "'N". Section normalisersect. Variable gT : finGroupType. Variables A B : {group gT}. Local Open Scope group_scope. Hypothesis nor : B \subset 'N(A). Lemma normaliser_equiv b : b \in B -> A \subset A :^ b. Proof. move=> bB. suff : A :^ b^-1 \subset A. by rewrite -sub_conjgV. move/subsetP : nor. move/(_ b^-1). move/groupVr in bB. move/(_ bB). rewrite /normaliser. rewrite inE. done. Qed. End normalisersect. (** There are many ways to state the fact that a subgroup is normal. For example, a subgroup $N$ of $G$ that satisfies $N x = x N$ for all $x \in G$ is a %\mydef{normal}% subgroup of $G$ (or is normal in $G$). We write %$N \trianglelefteq G$% if $N$ is normal in $G$. $H$ is normal in $G$ is noted %\lstinline!H <| G!% in %\mathcomp%, it is a boolean binary predicate (definition: %\lstinline!normal!%; notation scope: %\lstinline!group_scope!%; file: %{\tt fingroup.v}%). *) Print normal. (** The following example was originally taken from %\cite{map2012}% (%{\tt exercises-10.v}%). *) Section normalsect. Variable gT : finGroupType. Variables (H G : {group gT}). Local Open Scope group_scope. Hypothesis HG : H <| G. Lemma normal_commutes : H * G = G * H. Proof. case/normalP : HG => HG' nor. apply/setP => x. case/boolP : (_ \in G * _). case/mulsgP => x1 x2 Hx1 Hx2 x1x2. move: (nor _ Hx1). move/setP/(_ x2). rewrite Hx2. case/imsetP => h1 Hh1 x2x1. rewrite x1x2 x2x1. rewrite -conjgC. apply/mulsgP. by exists h1 x1. rewrite (mulGSid HG'). rewrite (mulSGid HG'). by move/negbTE. Qed. End normalsect.