Find all subsets of the octic group.

[amanda_ou812]

Homework Statement

G = {e, a, a^2, a^3, b, y, D (delta), T (theta)} Where e=(1), a=(1, 2, 3, 4), a^2 = (1, 3)(2, 4), a^3 = (1, 4, 3, 2), b = (1, 4)(2, 3), y = (2, 4), D = (1, 2)(3, 4), T = (1, 3)

Find the subsets.

Homework Equations

I know that the order of G is 8. So, my subsets should have an order of 1, 2, 4, or 8

The Attempt at a Solution

I am looking at other examples and I do not understand how they make the subgroups. Is it just guessing or is there a method.

By googling, I know that there are 10 subgroups. So my subgroups with order 1 are just {e}, subgroups with order 2 are {e, a^2} {e, b} {e , y} {e, D}, {e, T}

The next ones are where I get confused. I would think that my subgroups with order 4 would be {e, a} and {e, a^3} but internet resources say that it is {e, a, a^3} but why?

And, using {e, a, a^3}, i only get 8 subgroups (including G). What are the other two?

I suppose my general question is: is there a method?
Thanks!

 

[micromass]

Homework Statement

G = {e, a, a^2, a^3, b, y, D (delta), T (theta)} Where e=(1), a=(1, 2, 3, 4), a^2 = (1, 3)(2, 4), a^3 = (1, 4, 3, 2), b = (1, 4)(2, 3), y = (2, 4), D = (1, 2)(3, 4), T = (1, 3)

Find the subsets.

Homework Equations

I know that the order of G is 8. So, my subsets should have an order of 1, 2, 4, or 8

The Attempt at a Solution

I am looking at other examples and I do not understand how they make the subgroups. Is it just guessing or is there a method.

By googling, I know that there are 10 subgroups. So my subgroups with order 1 are just {e}, subgroups with order 2 are {e, a^2} {e, b} {e , y} {e, D}, {e, T}

The next ones are where I get confused. I would think that my subgroups with order 4 would be {e, a} and {e, a^3} but internet resources say that it is {e, a, a^3} but why?

And, using {e, a, a^3}, i only get 8 subgroups (including G). What are the other two?

I suppose my general question is: is there a method?
Thanks!

Hmm, you're the second one on this forum asking the exact same question...

Anyway, some remarks:
1) You probably mean subgroup instead of subset.
2) Neither {e, a}, {e,a³} or {e,a,a³} is a subgroup. The reason is that subgroups must be closed under the operation. So if a is in the subgroup, then so must a*a.
3) There is a method. First find all cyclic subgroups, then find the subgroups generated by two elements, then find the subgroups generated by three elements, and so on. This seems to be a lot of work, but it isn't if you're a bit smart.