Isomorphic group needed for cayley table

[jackscholar]

Homework Statement

I need to find an isomorphic group for the following group
F A B C D E F G H < these are the rotations/reflections, f is the operation followed by
A G D E B C H A F
B D G F A H C B E
C E F G H A B C D
D B A H G F E D C
E C H A F G D E B
F H C B E D G F A
G A B C D E F G H
H F E D C B A H G
^
These are again the rotations/reflections

Homework Equations

The Attempt at a Solution

Someone mentioned that there could be an isomorphism with multiplication under modulo 17 but with my limited knowledge in isomorphic groups I was unable to re-arrange the group as such. Any help would be highly appreciated

 

[Hurkyl]

(I bet they meant 16, not 17)

It's a group with 8 elements, right? There aren't very many of those. Have you at least found which one is the identity?

 

[jackscholar]

I haven't found a numerical Identity. I've just been looking at square values for numbers under 17 to see if there's a relationship but I haven't found anything

 

[Hurkyl]

Nono, I mean what is the identity element of your group? We can learn a lot by actually working in your group and determining a few simple properties.

 

[jackscholar]

G is the indentity and they all have self inverses

 

[Hurkyl]

Right. And since there is only one group with that property, your group has to be isomorphic to it!

Do you know what group that is? Hint:

Spoiler

it's abelian

The squares modulo 17 don't have this property: that group is a cyclic group. The units modulo 16 don't either (mistake on my part: I was thinking of the fact that the units modulo 8 have that property that they're all self-inverses).

 

[jackscholar]

Is there an isomorphic abelian group with 8 elements though. Thats what I need to find. I'm also unaccustomed to cyclic groups, our teacher made us skip it.