The Simplified Hasse diagram of A_5

 A5 is the alternating group on 5 elements. Since the order of S5 is 120, the order of A5 is half the size at 60. Here is the Simplified Hasse Diagram of the structure of the subgroups.

The red circles indicate the normal subgroups, which in this case are just the whole group and the identity. having just two normal subgroups means this is a simple group.

 

Red arrows can only originate in normal subgroups, and in this case, the identity is a normal subgroup of every subgroup of prime order.

 

There are 15 subgroups of order 2, all of the generators of the form (ab)(cd). 

 

There are 10 subgroups of order 3, all of generators of the form (abc).

The 6 subgroups of order 5 have generators of the form (abcde).

For the Klein-4 subgroups, we have the identity and three double transpositions that all keep the same element fixed. For example, (12)(34), (13)(24) and (14)(23) all leave 5 untouched. Likewise, the five copies of A4 all leave one element fixed.

The subgroups of order 5, 6 and 10 do not leave any element fixed, so they are not subgroups of A4. Of those three orders, only 6 divides 12, but the groups isomorphic to S3 are generated by a 3-cycle (abc) and a double transposition (ab)(de), so no element is fixed.

 

Every purple arrow goes from a subgroup of order n to a subgroup of order 2n. These subgroups are not normal, which is indicated by the blue rectangles, nor are they subnormal, since subnormality only occurs if a subgroup is part of a chain of subgroups, each normal in the next subgroup up in the chain and terminating in the entire group. A simple group that has any proper subgroup that is not the identity cannot contain a subnormal subgroup, so these subgroups that have purple arrows coming out of them are normalish, the phrase I am using until I learn the proper term.

 

The sum of all the numbers in red circles and blue rectangles is 59, the total number of subgroups.

 

The next Simplified Hasse diagram will be of S5, which will be a much larger undertaking with 156 total subgroups.

 

 

More Simplified Hasse Diagrams, S_3, A_4 and S_4.

 

Today's post is all about Simplified Hasse Diagrams. I didn't label this one to specify the group, but the only group of order 6 that is non-abelian is S3. In this group, subgroups are either normal, indicated by red ovals, on not normal, indicated by the blue rectangle.

The next diagram is for A4, order 12 and slightly more complex than S3. Notice the purple arrow that connects the order 2 subgroups to the Klein-4 group at order 4. The order 2 groups are subnormal, because the are normal in the next group up, not normal in the entire group, and there is a chain starting at any order 2 subgroup, up to the Klein-4 and ending at A4.

 

 

 

 

 

 

 

 

 

 

 

 

And then we have the Simplified Hasse diagram for S4, which is much more complicated than the preceding diagrams.

 

First complication: We have order 4 subgroups that are not isomorphic to one another, 1 Klein-4 and 3 copies of Z4.

 

Second complication: while all groups of order 2 are isomorphic, three are generated by even permutations, specifically (12)(34), (13)(23) and (14)(23), while six are generated by odd permutations, (12), (13), (14), (23), (24) and (34).

 

We now have four purple arrows. The even permutation subgroups of order 2 are still subnormal, starting a chain that goes through the Klein-4 subgroup, up to the A4 and on to the S4. But there are other purple arrows that lead to non-normal subgroups and no chain leads up to the entire group.

 

Specifically, the order 3 subgroups are normal in the order 6 subgroups, but the order 6 subgroups have both even and odd permutations, so they are not subgroups of A4. They are non-normal subgroups of S4.

 

Likewise, the Z4 subgroups are normal in their D4 supergroups, but the D4 subgroups are not normal is S4.

 

So we have the groups with a single generator, Z3 and Z4, normal in the groups just above them, S3 and D4 respectively, but not part of a subnormal chain. Until I can find out what the correct word is, I am calling these subgroups normalish. I assume there is a term because they can be found in very well known groups of small order.

 

The "culprits" that keep these subgroups from being subnormal are the subgroups that have both even and odd permutations in them. In S4, only the subgroups of all even permutations are normal.

 

Next up: A5 and S5, which are order 60 and 120 respectively. These will clearly be more convoluted.