Question on group theory: simplest math construction

Abolaban
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Hello Big Minds,

In the following analysis...It is said that D_2 contains three subgroups Z_2...why did he choose a mathematical constuction contains only two of the the three subgroups? shouldn't he use the three in his construction? what will happen if he used the three?

upload_2015-2-1_10-52-31.png

[from the book of P.Ramond, Group Theory, p8.]

best regards

Abolaban
 
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How many distinct elements would the direct product group have if he used 3 subgroups, each with 2 elements in them? If a direct product has 8 elements , it can't be isomorphic to a group with only 4 elements in it. His goal was to write an direct product that is essentially the same group as the 4 element group.
 

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