# Access Functions for Groups

The functions described here provide access to basic information stored for a polycyclic group $G$.

## `G . i: GrpGPC, RngIntElt -> GrpGPCElt`

The $i$-th polycyclic generator for $G$ if $i>0$, the inverse of the $|i|$-th polycyclic generator for $G$ if $i<0$ and the identity element of $G$ if $i=0$.

## `Generators(G): GrpGPC -> { @ GrpGPCElt @}`

## `PCGenerators(G): GrpGPC -> { @ GrpGPCElt  @}`

An indexed set containing the polycyclic generators of $G$.

## `Generators(H, G): GrpGPC, GrpGPC -> { @ GrpGPCElt @}`

## `PCGenerators(H, G): GrpGPC, GrpGPC -> { @ GrpGPCElt  @}`

An indexed set containing the polycyclic generators of $H$ as elements of $G$.

## `NumberOfGenerators(G): GrpGPC -> RngIntElt`

## `Ngens(G): GrpGPC -> RngIntElt`

## `NumberOfPCGenerators(G): GrpGPC -> RngIntElt`

## `NPCgens(G): GrpGPC -> RngIntElt`

## `NPCGenerators(G): GrpGPC -> RngIntElt`

The number of polycyclic generators for the polycyclic group $G$.

## `PCExponents(G): GrpGPC -> [RngIntElt]`

The orders of the cyclic factors in the polycyclic series defined by the polycyclic presentation of $G$. The orders are returned in a sequence $Q$. $|G_i/G_{i+1}| = m_i = Q[i]$ if $Q[i] > 0$ and $G_i/G_{i+1}$ is infinite (i.e. $i\notin I$) if $Q[i] = 0$.

## `HirschNumber(G): GrpGPC -> RngIntElt`

The Hirsch number of $G$, i.e. the number of infinite cyclic factors in the polycyclic series defined by the polycyclic presentation of $G$.

The Hirsch number of $G$ is equal to $n - |I|$, i.e. to the number of polycyclic generators of $G$ for which there is no power relation. A polycyclic group $G$ is finite if and only if its Hirsch number is 0.