I'm reading Nathan Carter's "Visual Group Theory" and I'm a little stuck on his choice of words and want to make sure I understand his statement correctly. It concerns normal subgroups and is stated as follows:
" ...at Figure 6.8 on page 104, which helped us picture a left coset gH as the copy of H to which the g arrow from e leads. That figure also depicts the right coset Hg as the nodes that g arrows can reach from H. These visualizations tell us that the defining characteristic of normal subgroups, gH = Hg, can be restated as follows: To whichever coset one g arrow leads from H (the left coset gH), all g arrows lead unanimously (because it is also the right coset Hg)."
I'm confused about the part "the left coset gH" in parentheses. If gH is the copy of H at the node g, what does this have to do with "whichever coset one g arrow leads from H"?
For a general subgroup H, the left cosets gH look like copies of H at the node g in the Caley diagram (we could say that left multiplication by g moves H as a block), while right cosets are more scattered, in the sense that different elements of the right coset belong to different left cosets. But when H is normal (gH=Hg), since every right coset is also a left coset, right multiplication must also move H as a block around the Caley diagram, and that is probably the intended meaning of "all g arrows lead unanimously".
So I think the intended meaning of the sentence is: The defining characteristic of a normal subgroup is that all g arrows from H lead to the same left coset, which must be gH (since gH and Hg always share at least one element: g)
So: "To whichever coset one g arrow leads from H (the left coset gH [NOTE: Because in fact every element of Hg must belong to some left coset]) all g arrows lead unanimously (because it is also the right coset Hg [NOTE: Because gH=Hg])"