- #1
nhartung
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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, a2, b, b2, c, c2, d, d2}
G is isomorphic to Z/3 x Z/3
a3 = e
b3 = e
c3 = e
d3 = e
Now i'll show the mappings of G onto Z/3 x Z/3:
e -> (0,0)
a -> (1,0)
a2 -> (2,0)
b -> (0,1)
b2 -> (0,2)
c -> (1,1)
c2 -> (2,2)
d -> (1,2)
d2 -> (2,1)
Did I do everything correctly here, and is this sufficient to find all groups of order 9 as the problem is asking?