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## Homework Statement

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

## Homework Equations

None

## 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?