## Isomorphisms of Groups

Two groupsandare isomorphic if they have the same group structure, so the the elements of one group can be put in a one to one correspondence with the elements of the other. All the relationships between the elements of each group are preserved by the correspondence. If we can label the mapping of one group to the other asthen

1. The identity of the first groupis mapped byto the identity of the second,so that

2. The order of each group is the same:

3 Ifwith group operationandwith group operation then

4. The proprty of being cyclic or abelian is preserved by an ismorphism, as is the order if each element, and the property of being inverse. All of these are necessary condictions to preserve the group structure.

5. The Group or Cayley table for each group must be the same size, and must be capable of being arranged so that both tables have the same arrangement of elements.

Example: The groupsandare isomorphic. The Cayley tables are given below.

0 | 1 | 2 | 3 | |

0 | 0 | 1 | 2 | 3 |

1 | 1 | 2 | 3 | 0 |

2 | 2 | 3 | 0 | 1 |

3 | 3 | 0 | 1 | 2 |

1 | -1 | i | -i | |

1 | 1 | -1 | i | -i |

-1 | -1 | 1 | -i | i |

i | i | -i | -1 | 1 |

-i | -i | i | 1 | -1 |

The tables do not have the same structure.

0 | 1 | 2 | 3 | |

0 | 0 | 1 | 2 | 3 |

1 | 1 | 2 | 3 | 0 |

2 | 2 | 3 | 0 | 1 |

3 | 3 | 0 | 1 | 2 |

1 | -1 | i | -i | |

1 | 1 | -1 | i | -i |

-1 | -1 | 1 | -i | i |

i | i | -i | -1 | 1 |

-i | -i | i | 1 | -1 |

All the highlighted elements in the first table are the same, but in the second table they are different. We can use the mapping

and(1)

This means swapping the elements -1 and i throughout the second table. We obtain

0 | 1 | 2 | 3 | |

0 | 0 | 1 | 2 | 3 |

1 | 1 | 2 | 3 | 0 |

2 | 2 | 3 | 0 | 1 |

3 | 3 | 0 | 1 | 2 |

1 | i | -1 | -i | |

1 | 1 | i | -1 | -i |

i | i | -1 | -i | 1 |

-1 | -1 | -i | 1 | i |

-i | -i | 1 | i | -1 |