# Quotient Groups

## Construction of Quotient Groups

One of the strengths of representing groups with polycyclic or power-conjugate presentations is that arbitrary quotient groups can be computed. Given (generators for) a normal subgroup of a pc-group, Magma will compute a pc-presentation for the quotient and the corresponding canonical homomorphism.

The `pQuotient` function, which can be used to find a prime-power quotient of a finitely-presented group, can also be used to compute quotients of pc-groups.

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

Construct the quotient $Q$ of the pc-group $G$ by the normal subgroup $N$, where $N$ is the smallest normal subgroup of $G$ containing the elements specified by the terms of the *generator list* $L$.

The possible forms of a term $L[i]$ of the generator list are the same as for the `sub`-constructor.

The quotient group $Q$ and the corresponding natural homomorphism $f : G \rightarrow Q$ are returned.

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

Given a normal subgroup $N$ of the pc-group $G$, construct the quotient of $G$ by $N$.

### `Example: Pc Quotient (ex-934992)`

We will compute $O_{3',3}(G)$, where $G$ is a pc-representation of the symmetric group $S_4$. The subgroup is defined by $O_{3',3}(G)/O_{3'}(G) = O_3(G/O_{3'}(G))$.

```magma
> G := PCGroup(Sym(4));
> N := pCore(G,-3);
> Q,f := quo<G|N>;
> Q;
GrpPC : Q of order 6 = 2 * 3
PC-Relations:
    Q.1^2 = Id(Q),
    Q.2^3 = Id(Q),
    Q.2^Q.1 = Q.2^2
> S := pCore(Q,3);
> H := S @@ f;
> H;
GrpPC : H of order 12 = 2^2 * 3
PC-Relations:
    H.1^3 = Id(H),
    H.2^2 = Id(H),
    H.3^2 = Id(H),
    H.2^H.1 = H.2 * H.3,
    H.3^H.1 = H.2

```

## Abelian and $p$-Quotients

A number of standard quotients may be constructed.

### `AbelianQuotient(G): GrpPC -> 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.

### `AbelianQuotientInvariants(G): GrpPC -> SeqEnum`

### `AQInvariants(G): GrpPC -> SeqEnum`

A sequence of integers giving the abelian invariants of the maximal abelian quotient of $G$.

### `ElementaryAbelianQuotient(G, p): GrpPC, 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 : parameters): GrpPC, RngIntElt, RngIntElt -> GrpPC, Map`

```magma
Workspace : RngIntElt                    Default: 5000000
Metabelian: BoolElt                      Default: false
Exponent  : RngIntElt                    Default: 0
Print     : RngIntElt                    Default: 0
```

Given a pc-group $G$, a prime $p$, and a positive integer $c$, this function constructs a consistent power-conjugate presentation for the largest $p$-quotient $P$ of $G$ having lower exponent-$p$ class at most $c$. If $c$ is given as zero, 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 $P.r^{P.s}$.