# Thread: Abstract Algebra: Groups and Symmetry

1. ## Abstract Algebra: Groups and Symmetry

Hi all,

Again, this is a homework problem, so please don't give me a full solution.

I have not tried this problem as yet, but I am posting ahead of time, just in case i will need help with it. It seems pretty straight forward from the wording, but oh, well, you never know, i'm stupid sometimes...

if no one replies, i'll tell you if i get it or not. i won't look at the hints given until i try the problem myself (after i read the chapter in the book).

Here goes, short and sweet

Problem:

Determine the group symmetries of a regular pentagon. (It will have order 10.)

As i said, i have not read the chapter, so i have no background info for you guys.

Thanks

2. Originally Posted by Jhevon
Hi all,

Again, this is a homework problem, so please don't give me a full solution.

I have not tried this problem as yet, but I am posting ahead of time, just in case i will need help with it. It seems pretty straight forward from the wording, but oh, well, you never know, i'm stupid sometimes...

if no one replies, i'll tell you if i get it or not. i won't look at the hints given until i try the problem myself (after i read the chapter in the book).

Here goes, short and sweet

Problem:

Determine the group symmetries of a regular pentagon. (It will have order 10.)

As i said, i have not read the chapter, so i have no background info for you guys.

Thanks
I don't know how you are expected to do this but the elements are the 5 rotations by $\frac{2 \pi n}{5},\ n=1,\ 2,\ 3,\ 4,\ 5$, and the 5 reflections about axes through the centre of the pentagon and a vertex.

RonL

3. Originally Posted by CaptainBlack
I don't know how you are expected to do this but the elements are the 5 rotations by $\frac{2 \pi n}{5},\ n=1,\ 2,\ 3,\ 4,\ 5$, and the 5 reflections about axes through the centre of the pentagon and a vertex.

RonL
hehe, i just finished reading the chapter and that's what i thought. i believe that's right, since they say it is of order 10. but i noticed this thing in the book about reflection in a horizontal line and a vertical line. i believe only one of these can work at a time because of the way the pentagon is shaped. that is, if i imagine it in space with the base horizontal, then one symmetry is a reflection in the line that passes through the top vertex but bisects the base. do we count this? or should we only reflect in lines going through the vertices?

EDIT: oh wait! i see. this is counted in the reflections about the axis... thanks

4. This is the dihedral group. There are 5 rotations of the pentagon. And then the 5 reflections give us 10.