Changes

|}

|}

In many cases, one finds that both groups are written with the same sign of operation, typically "<math>\cdot</math>", "+", "*", or simple concatenation, but they remain in general distinct whether considered as operations or as relations, no matter what signs of operation are used.  In such a setting, our chiasmatic theme may run a bit like these two variants:

In many cases, one finds that both group operations are indicated by the same sign, typically &nbsp;<math>\cdot\!</math>&nbsp;, &nbsp;<math>*\!</math>&nbsp;, &nbsp;<math>+\!</math>&nbsp;, or simple concatenation, but they remain in general distinct whether considered as operations or as relations, no matter what signs of operation are used.  In such a setting, our chiasmatic theme may run a bit like these two variants:

: The image of the sum is the sum of the images.

| <p>The image of the sum is the sum of the images.</p>

Figure&nbsp;22 presents a generic picture for groups <math>G\!</math> and <math>H.\!</math>

 

Figure 22 presents a generic picture for groups ''G'' and ''H''.

<pre>

<pre>

o-----------------------------------------------------------o

o-----------------------------------------------------------o

|                                                          |

|                                                          |

|                      G          H                      |

|                      G          H                      |

|                      @           @                       |

|                      O           O                       |

|                      /|\        /|\                      |

|                      /|\        /|\                      |

|                    / | \      / | \                    |

|                    / | \      / | \                    |

|                      \  \  \  /                      |

|                      \  \  \  /                      |

|                        \ / \ / \ /                        |

|                        \ / \ / \ /                        |

|                        @   @   @                         |

|                        O   O   O                         |

|                        J  J  J                        |

|                        J  J  J                        |

|                                                          |

|                                                          |

Figure 22.  Group Homomorphism J : G <- H

Figure 22.  Group Homomorphism J : G <- H

</pre>

</pre>

In a setting where both groups are written with a plus sign, perhaps even constituting the very same group, the defining formula of a morphism, ''J''(''L''(''u'',&nbsp;''v'')) = ''K''(''Ju'',&nbsp;''Jv''), takes on the shape ''J''(''u''&nbsp;+&nbsp;''v'') = ''Ju''&nbsp;+&nbsp;''Jv'', which looks very analogous to the'distributive multiplication of a sum (''u''&nbsp;+&nbsp;''v'') by a factor ''J''.  Hence another popular name for a morphism:  a "linear" map.

In a setting where both groups are written with a plus sign, perhaps even constituting the very same group, the defining formula of a morphism, <math>J(L(u, v)) = K(Ju, Jv),\!</math> takes on the shape <math>J(u + v) = Ju + Jv,\!</math> which looks very analogous to the distributive multiplication of a sum <math>(u + v)\!</math> by a factor <math>J.\!</math> Hence another popular name for a morphism:  a ''linear'' map.

===Commentary Note 11.16===

===Commentary Note 11.16===