# Isomorphic group needed for cayley table

• jackscholar
In summary, 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.
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

## 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

(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?

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

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.

G is the indentity and they all have self inverses

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:
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).

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.

## 1. What is an isomorphic group?

An isomorphic group is a mathematical concept that refers to two groups being structurally identical. This means that the groups have the same elements, the same operation, and the same structure, but the elements may be labeled differently.

## 2. Why is an isomorphic group needed for a Cayley table?

A Cayley table is a visual representation of a group's operation. By using an isomorphic group, we can simplify the table and make it easier to see patterns and relationships within the group.

## 3. Can any group have an isomorphic group?

Yes, any group can have an isomorphic group. Two groups are isomorphic if they have the same number of elements and the same group operation.

## 4. How do you determine if two groups are isomorphic?

To determine if two groups are isomorphic, you can compare their Cayley tables. If the tables have the same structure and the elements are labeled differently, then the groups are isomorphic. Another way is to find a function that maps one group onto the other, preserving the group operation.

## 5. What is the significance of an isomorphic group in mathematics?

An isomorphic group is significant because it allows for easier visualization and understanding of a group's structure and operation. It also helps mathematicians to study and classify different types of groups by identifying isomorphic groups within a larger set of groups.

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