(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find all groups of order 9, order 10, order 11.

2. Relevant equations

None

3. The attempt at a solution

We have already done an example in class of groups of order 4 and of order 2,3,5, or 7.

So i'm going to base my proofs on the example of groups of order 4 except for the group of order 11 which I suspect is acting like the groups of order 2,3,5 or 7 since it is also a prime number.

Here is my attempt at groups of order 9, i'm a little unsure about the final part.

Let G be a group of order 9, every element has order 1, 3, or 9. If there is an element g of order 9, then <g> = G. G is cyclic and isomorphic to (Z/9, +).

If there is no element of order 9, the (non-identity) elements must all have order 3.

G = {e, a, a^{2}, b, b^{2}, c, c^{2}, d, d^{2}}

G is isomorphic to Z/3 x Z/3

a^{3}= e

b^{3}= e

c^{3}= e

d^{3}= e

Now i'll show the mappings of G onto Z/3 x Z/3:

e -> (0,0)

a -> (1,0)

a^{2}-> (2,0)

b -> (0,1)

b^{2}-> (0,2)

c -> (1,1)

c^{2}-> (2,2)

d -> (1,2)

d^{2}-> (2,1)

Did I do everything correctly here, and is this sufficient to find all groups of order 9 as the problem is asking?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Find all groups of order 9, order 10, and order 11

**Physics Forums | Science Articles, Homework Help, Discussion**