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To list all the finite group up to order 9, we use the following results:
Theorem. A finite group of a prime number order is cyclic.
Consequently there is only one group of each of the orders , namely the cyclic groups ,, and .
The groups of order 4 and 9 are covered by the theorem:
Theorem. Let the order of a group is , where is a prime number. Then is Abelian. Moreover either is cyclic of order , or it is a direct product of two cyclic groups of order .
This result implies that there are two groups of order 4 and 9. These are groups cyclic groups or , or , and . Each of them is Abelian.
The group is also known as the Klein four group .
Let be a group of order 8. Lagrange's Theorem implies that the order of an element of divides the order of , that is 8. Therefore each element of has order 1,2,4 or 8.
If there is an element of order 8 then is cyclic, i.e. .
If contains no element of order 4 then must consist of elements with order 2 (and the identity). It is easy to see that such a group is abelian. Namely, in this case , i.e. for every . Then for every .
So for to be non-Abelian it must have an element of order 4, say . This gives us 4 elements of : . If there is an element different from one of these with order 2, say, then since also has order 4 and the group generated by is a normal subgroup (since it has only 2 cosets). This implies:
Consequently . Since and , the equalities and are impossible. In both cases , and since , would be Abelian. Therefore it remains that and and the resulting group is the Dihedral group .
The group is generated by two elements and , where is of order 4, of order 2, such that .
So this leaves us with the case that all the elements of not belonging to must have order 4. Let be one of these. Since has order 2, it must equal . The element cannot be a power of or . So again has order 4. The elements of are therefore . This group is isomorphic to the Quaternion group .
The group is generated by two elements and , both of order 4 such that and . When using quaternion unities to describe , the isomorphism is given by mapping to and to .
For groups of order 6 we have two possibilities: the cyclic group or the direct product of two cyclic groups .
Cite this web-page as:
Štefan Porubský: Finite groups up to order 9.