# Determine the subgroup lattice for Z8

1. Sep 11, 2012

### srfriggen

1. The problem statement, all variables and given/known data

"Determine the subgroup lattice for Z8"

2. Relevant 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}

3. 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!

Last edited: Sep 11, 2012