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Calculating quotient groups and normal subgroups

Posted 2 years ago

Hello! I am trying to calculate all the quotient groups of the D4 group. I'm using the following Mathematica function to do so:

Q = FiniteGroupData[Group, "QuotientGroups"]

Which produces

{"Trivial", {"CyclicGroup", 2}, {"DihedralGroup", 
  2}, {"DihedralGroup", 4}}

I found the Normal subgroups as well, by doing

r = FiniteGroupData[Group, "NormalSubgroups"];
r = DeleteDuplicates[r, 
  IsomorphicGraphQ[FiniteGroupData[#1, "CycleGraph"], 
    FiniteGroupData[#2, "CycleGraph"]] &]

and got

{{"CyclicGroup", 1}, {"CyclicGroup", 2}, {"CyclicGroup", 
  4}, {"DihedralGroup", 2}, {"DihedralGroup", 4}}

According to the documentation centre the possible quotient groups, are obtained by finding the quotient with respect to normal subgroups, my question is if there's a way of knowing which normal subgroup was used to get each quotient group. For example, is there a way to know that the normal subgroup used to get the quotient group D2 was C2? I haven't found any way of doing this and could use some help, please.

Thanks!

I didn't find any algorithm is able to find the normal subgroup corresponding to a quotient group.

I think there is a obstacle, which is the normal subgroups and quotient groups do not correspond to each other one by one, i.e. the map is not one-one and onto. For example:

As you can see in the notebook above, D4 group have 5 normal subgroups but only 4 quotient groups, which means that there are two normal subgroups correspond to a quotient group together, or two normal subgroups are isomorphic to each other. According to the law of group order, only D2 and C4 could be and they are not isomorphic to each other, of course. But they generate the same quotient group C2.

POSTED BY: Bowen Ping
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