# Isomorphism

We provide two functions dealing with isomorphisms of $E$-groups. One for deciding isomorphism between two $E$-groups, and the other for constructing a generating set for the automorphism group of such groups. These algorithms are based on (Theorem E) Maglione–Stanojkovski.

## EGAutomorphismGroup

**Input**:

`GrpPC`

$G$.

**Output**:

`GrpAuto`

$A$.

Return $A=\mathrm{Aut}(G)$ provided $G$ is an $E$-group.

As we record in Maglione–Stanojkovski, the theoretical performance of `EGAutomorphismGroup`

is $O(|G|^{1/9})$, and the actual performance can be observed in the following scatter plot of data.

## EGIsIsomorphic

**Input**:

`GrpPC`

$G$,`GrpPC`

$H$.

**Output**:

`BoolElt`

,`Map`

$\varphi$.

Decide if $G$ and $H$ are $E$-groups and if $G\cong H$, and if so return an isomorphism $\varphi : G \to H$.