Determine the subgroup lattice for Z8

[srfriggen]

Homework Statement

"Determine the subgroup lattice for Z₈"

Homework Equations

<1>={1,2,3,4,5,6,7,0}
<2>={2,4,6,0}
<3>=<1>=<5>=<7>
<4>={4,0}
<6>={6,4,2,0}

The Attempt at a Solution

My book only mentions this topic in one sentence and shows a diagram for Z₃₀, which looks like a cube.

I don't quite see all the logic behind it. One "track" is <1> -- <3> -- <6> -- <0>. It's easy to see that each of these are subgroups of one another, but why wouldn't <1> be connected directly to <0> as well?

My classmate told me he thinks it should look like a tetrahedron on edge.

I'm coming up with an square based pyramid sitting upside down, with <0> as the "top" of the pyramid (but upside down). I guess it doesn't matter how it really looks. Here are the paths I have, all stemming from <1>

One chain looks like <1> - <2> - <4> - <0>
another looks liks <1> - <6> - <4> - <0>
the third is <1> - <6> - <0>
and the last is <1> - <2> - <0>
Any advice on how to proceed would be greatly appreciated.

Thanks!

 

[mighty2000]

Hello,

It seems like you have already made some good progress in determining the subgroup lattice for Z8. The key concept to understand here is that each subgroup must be a subset of the group Z8 and must also be closed under the group operation (in this case, addition). This means that each subgroup must contain the identity element (0) and must also contain all elements that can be obtained by adding any two elements in the subgroup.

Your first "track" of <1> - <3> - <6> - <0> is correct. This is because <3> and <6> are subgroups of <1> and <0> is the identity element that is shared by all subgroups.

For your question about why <1> is not connected directly to <0>, this is because <0> is not a proper subgroup of <1>. It is the identity element that is shared by all subgroups, but it does not meet the criteria of being a subset of <1> and being closed under the group operation.

Your classmate's suggestion of a tetrahedron on edge is also a valid representation of the subgroup lattice for Z8. This is because each edge represents a subgroup and each vertex represents the identity element (0).

Your square based pyramid is also a valid representation, with <0> at the top and the other subgroups branching out from it. As long as all the subgroups are connected and follow the rules of being a subset and closed under the group operation, the exact shape of the lattice may vary.

To continue, you can try to find all possible subgroups of Z8 and connect them in the lattice. Remember to check if each subgroup meets the criteria and if it is already included in the lattice. You can also try to find any relationships between the subgroups, such as which subgroups are proper subgroups of others.

I hope this helps and good luck with your exploration of the subgroup lattice for Z8!