Section 13.3 Sage
¶Cyclic groups, and direct products of cyclic groups, are implemented in Sage as permutation groups. However, these groups quickly become very unwieldly representations and it should be easier to work with finite abelian groups in Sage. So we will postpone any specifics for this chapter until that happens. However, now that we understand the notion of isomorphic groups and the structure of finite abelian groups, we can return to our quest to classify all of the groups with order less than \(16\text{.}\)
Subsection Classification of Finite Groups
It does not take any sophisticated tools to understand groups of order \(2p\text{,}\) where \(p\) is an odd prime. There are two possibilities — a cyclic group of order \(2p\) and the dihedral group of order \(2p\) that is the set of symmetries of a regular \(p\)gon. The proof requires some close, tight reasoning, but the required theorems are generally just concern orders of elements, Lagrange's Theorem and cosets. See Exercise 9.3.55. This takes care of orders \(n=6,\,10,\,14\text{.}\)
For \(n=9\text{,}\) the upcoming Corollary 14.16 will tell us that any group of order \(p^2\) (where \(p\) is a prime) is abelian. So we know from this section that the only two possibilities are \({\mathbb Z}_9\) and \({\mathbb Z}_3\times{\mathbb Z}_3\text{.}\) Similarly, the upcoming Theorem 15.10 will tell us that every group of order \(n=15\) is abelian. Now this leaves just one possibility for this order: \({\mathbb Z}_3\times{\mathbb Z}_5\cong{\mathbb Z}_{15}\text{.}\)
We have just two orders left to analyze: \(n=8\) and \(n=12\text{.}\) The possibilities are groups we already know, with one exception. However, the analysis that these are the only possibilities is more complicated, and will not be pursued now, nor in the next few chapters. Notice that \(n=16\) is more complicated still, with \(14\) different possibilities (which explains why we stopped here).
For \(n=8\) there are \(3\) abelian groups, and the two nonabelian groups are the dihedral group (symmetries of a square) and the quaternions.
For \(n=12\) there are \(2\) abelian groups, and \(3\) nonabelian groups. We know two of the nonabelian groups as a dihedral group, and the alternating group on \(4\) symbols (which is also the symmetries of a tetrahedron). The third nonabelian group is an example of a “dicyclic” group, which is an infinite family of groups, each with order divisible by \(4\text{.}\) The order \(12\) dicyclic group can also be constructed as a “semidirect product” of two cyclic groups — this is a construction worth knowing as you pursue further study of group theory. The order \(8\) dicyclic group is also the quaternions and more generally, the dicyclic groups of order \(2^k\text{,}\) \(k>2\) are known as “generalized quaternion groups.”
The following examples will show you how to construct some of these groups, while also exercising a few of the commands and allowing us to be more certain the following table is accurate.
Subsection Groups of Small Order as Permutation Groups
We list here constructions, as permutation groups in Sage, for all of the groups of order less than \(16\text{.}\)
Order  Construction  Notes, Alternatives 
1  CyclicPermutationGroup(1) 
Trivial 
2  CyclicPermutationGroup(2) 
SymmetricGroup(2) 
3  CyclicPermutationGroup(3) 
Prime order 
4  CyclicPermutationGroup(4) 
Cyclic 
4  KleinFourGroup() 
Abelian, noncyclic 
5  CyclicPermutationGroup(5) 
Prime order 
6  CyclicPermutationGroup(6) 
Cyclic 
6  SymmetricGroup(3) 
Nonabelian 
DihedralGroup(3) 

7  CyclicPermutationGroup(7) 
Prime order 
8  CyclicPermutationGroup(8) 
Cyclic 
8  C2=CyclicPermutationGroup(2) 

C4=CyclicPermutationGroup(4) 

G=direct_product_permgroups([C2,C4]) 
Abelian, noncyclic  
8  C2=CyclicPermutationGroup(2) 

G=direct_product_permgroups([C2,C2,C2]) 
Abelian, noncyclic  
8  DihedralGroup(4) 
Nonabelian 
8  QuaternionGroup() 
Quaternions 
DiCyclicGroup(2) 

9  CyclicPermutationGroup(9) 
Cyclic 
9  C3=CyclicPermutationGroup(3) 

G=direct_product_permgroups([C3,C3]) 
Abelian, noncyclic  
10  CyclicPermutationGroup(10) 
Cyclic 
10  DihedralGroup(5) 
Nonabelian 
11  CyclicPermutationGroup(11) 
Prime order 
12  CyclicPermutationGroup(12) 
Cyclic 
12  C2=CyclicPermutationGroup(2) 

C6=CyclicPermutationGroup(6) 

G=direct_product_permgroups([C2,C6]) 
Abelian, noncyclic  
12  DihedralGroup(6) 
Nonabelian 
12  AlternatingGroup(4) 
Nonabelian 
Symmetries of tetrahedron  
12  DiCyclicGroup(3) 
Nonabelian 
Semidirect product \(Z_3\rtimes Z_4\)  
13  CyclicPermutationGroup(13) 
Prime order 
14  CyclicPermutationGroup(14) 
Cyclic 
14  DihedralGroup(7) 
Nonabelian 
15  CyclicPermutationGroup(15) 
Cyclic 