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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.