# Quotient Groups

## Construction of Quotient Groups

### `quo<G | L>: GrpPerm, List -> GrpPerm, Map`

Given the permutation group $G$, construct the quotient group $Q = G/N$, where $N$ is the normal closure of the subgroup of $G$ generated by the elements specified by $L$. The clause $L$ is a list of one or more items of the following types:

**(a)**
A sequence of $n$ integers defining a permutation of $G$;

**(b)**
A set or sequence of sequences of type (a);

**(c)**
An element of $G$;

**(d)**
A set or sequence of elements of $G$;

**(e)**
A subgroup of $G$;

**(f)**
A set or sequence of subgroups of $G$.

Each element or group specified by the list must belong to the *same* generic permutation group. The function returns

**(a)**
the quotient group $Q$, and

**(b)**
the natural homomorphism $f: G\rightarrow Q$.

Currently, the quotient group is constructed via the regular representation of the quotient, so the application of this operator is restricted to the case where the index of $N$ in $G$ is small. The representation of the quotient group that is returned is the result of a degree reduction applied to the regular representation, so need not be regular. The generators of the quotient are images of the generators of $G$.

The second return value is the epimorphism from G to the resulting quotient group.

### `G / N: GrpPerm, GrpPerm -> GrpPerm`

Given a normal subgroup $N$ of the permutation group $G$, construct the quotient of $G$ by $N$. Currently, the quotient group is constructed via the regular representation of the quotient, so the application of this operator is restricted to the case where the index of $N$ in $G$ is small. The representation of the quotient group that is returned is the result of a degree reduction applied to the regular representation, so need not be regular. The generators of the quotient are images of the generators of $G$.

### `Example: Quotient (ex-d39a4a)`

The quotient of ${\operatorname{Sym}}(4)$ by the Klein $4$-group is constructed by the following statement:

```magma
> Q, f := quo< Sym(4) | (1,2)(3,4), (1,3)(2,4) >;
> Q;
Permutation group Q acting on a set of cardinality 3
Order = 6 = 2 * 3
    (2, 3)
    (1, 2)

```

## Abelian, Nilpotent and Soluble Quotients

A number of standard quotients may be constructed. The method first constructs a presentation for the permutation group and then applies the appropriate fp-group algorithm.

### `AbelianQuotient(G): GrpPerm -> GrpAb, Map`

The maximal abelian quotient $G/G^\prime$ of the group $G$ as `GrpAb` (cf. Chapter [Abelian Groups](../../FinitelyPresentedGroups/AbelianGroups/index-abelian-groups.md#chapgrpab)). The natural epimorphism $\pi:G\rightarrow G/G^\prime$ is returned as second value.

### `ElementaryAbelianQuotient(G, p): GrpPerm, RngIntElt -> GrpAb, Map`

The maximal $p$-elementary abelian quotient $Q$ of the group $G$ as `GrpAb` (cf. Chapter [Abelian Groups](../../FinitelyPresentedGroups/AbelianGroups/index-abelian-groups.md#chapgrpab)). The natural epimorphism $\pi:G\rightarrow Q$ is returned as second value.

### `pQuotient(G, p, c): GrpPerm, RngIntElt, RngIntElt -> GrpPC, Map, SeqEnum, BoolElt`

Given a permutation group $G$, a prime $p$ and a positive integer $c$, construct a pc-presentation for the largest $p$-quotient $P$ of $G$ having lower exponent-$p$ class at most $c$. If $c$ is given as $0$, then the limit 127 is placed on the class.

The function also returns the natural homomorphism $\pi$ from $G$ to $P$, a sequence $S$ describing the definitions of the pc-generators of $P$ and a flag indicating whether $P$ is the maximal $p$-quotient of $G$.

The $k$-th element of $S$ is a sequence of two integers, describing the definition of the $k$-th pc-generator $P.k$ of $P$ as follows.

- If $S[k] = [0,r]$, then $P.k$ is defined via the image of $G.r$ under $\pi$.

- If $S[k] = [r,0]$, then $P.k$ is defined via the power relation for $P.r$.

- If $S[k] = [r,s]$, then $P.k$ is defined via the conjugate relation involving $P.r^{P.s}$.

### `NilpotentQuotient(G, c): GrpPerm, RngIntElt -> GrpGPC, Map`

This function returns the class $c$ nilpotent quotient of $G$, together with the epimorphism $\pi$ from $G$ onto this quotient.

### `SolvableQuotient(G): GrpPerm -> GrpPC, Map, SeqEnum, MonStgElt`

### `SolubleQuotient(G): GrpPerm -> GrpPC, Map, SeqEnum, MonStgElt`

The function returns the largest soluble quotient $S$ of the permutation group $G$ together with the epimorphism $\pi:G\rightarrow S$.

### `Example: Special Quotient (ex-d5d186)`

The soluble quotient of the wreath product of ${\operatorname{Sym}}(6)$ with the dihedral group of order 12 is easily constructed:

```magma
> G := WreathProduct( Sym(6), DihedralGroup(6));
> #G;
1671768834048000000
> SQ, phi := SolubleQuotient(G);
SQ;
GrpPC : SQ of order 768 = 2^8 * 3
PC-Relations:
    SQ.1^2 = SQ.5,
    SQ.2^2 = Id(SQ),
    SQ.3^2 = Id(SQ),
    SQ.4^2 = Id(SQ),
    SQ.5^3 = Id(SQ),
    SQ.6^2 = Id(SQ),
    SQ.7^2 = Id(SQ),
    SQ.8^2 = Id(SQ),
    SQ.9^2 = Id(SQ),
    SQ.2^SQ.1 = SQ.2 * SQ.5,
    SQ.3^SQ.1 = SQ.3 * SQ.4 * SQ.6 * SQ.8,
    SQ.4^SQ.1 = SQ.4 * SQ.9,
    SQ.4^SQ.2 = SQ.4 * SQ.6 * SQ.7 * SQ.8,
    SQ.5^SQ.2 = SQ.5^2,
    SQ.5^SQ.3 = SQ.5 * SQ.7,
    SQ.5^SQ.4 = SQ.5 * SQ.6 * SQ.8,
    SQ.6^SQ.1 = SQ.6 * SQ.8,
    SQ.6^SQ.2 = SQ.7 * SQ.8,
    SQ.6^SQ.5 = SQ.6 * SQ.7 * SQ.8 * SQ.9,
    SQ.7^SQ.1 = SQ.8,
    SQ.7^SQ.2 = SQ.9,
    SQ.7^SQ.5 = SQ.7 * SQ.9,
    SQ.8^SQ.1 = SQ.7 * SQ.9,
    SQ.8^SQ.2 = SQ.6 * SQ.9,
    SQ.8^SQ.5 = SQ.6 * SQ.9,
    SQ.9^SQ.1 = SQ.6 * SQ.8 * SQ.9,
    SQ.9^SQ.2 = SQ.7,
    SQ.9^SQ.5 = SQ.7

```