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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:
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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:
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: The image of the sum is the sum of the images.
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{| align="center" cellspacing="6" width="90%"
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| <p>The image of the sum is the sum of the images.</p>
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|-
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| <p>The image of the product is the sum of the images.</p>
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: The image of the product is the product of the images.
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Figure&nbsp;22 presents a generic picture for groups <math>G\!</math> and <math>H.\!</math>
 
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Figure 22 presents a generic picture for groups ''G'' and ''H''.
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{| align="center" cellspacing="6" width="90%"
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| align="center" |
 
<pre>
 
<pre>
 
o-----------------------------------------------------------o
 
o-----------------------------------------------------------o
 
|                                                          |
 
|                                                          |
 
|                      G          H                      |
 
|                      G          H                      |
|                      @           @                       |
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|                      O           O                       |
 
|                      /|\        /|\                      |
 
|                      /|\        /|\                      |
 
|                    / | \      / | \                    |
 
|                    / | \      / | \                    |
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|                      \  \  \  /                      |
 
|                      \  \  \  /                      |
 
|                        \ / \ / \ /                        |
 
|                        \ / \ / \ /                        |
|                        @   @   @                         |
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|                        O   O   O                         |
 
|                        J  J  J                        |
 
|                        J  J  J                        |
 
|                                                          |
 
|                                                          |
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Figure 22.  Group Homomorphism J : G <- H
 
Figure 22.  Group Homomorphism J : G <- H
 
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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.
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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===
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