# Subgroups of Order Three

1. May 23, 2008

### e(ho0n3

[SOLVED] Subgroups of Order Three

The problem statement, all variables and given/known data
Let G be a subgroup containing exactly eight elements of order three. How many subgroups of order three does G have?

The attempt at a solution
This problem was discussed in class today. The professor said that G has four subgroups of order three. I didn't follow his explanation very well so I didn't understand why. Since there are eight elements of order three, wouldn't each of these elements constitute a subgroup of order three so G has at least eight subgroups of order three?

2. May 23, 2008

### DavidWhitbeck

Suppose $$a \in G$$ is order 3 then $$a^2$$ is also order 3. They belong to the same subgroup. That means that only 4 of the 8 are generators, and the other 4 are their squares.

3. May 23, 2008

### e(ho0n3

I see. I overlooked that fact. Thanks a lot.