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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 \(16\).

SubsectionClassification of Finite Groups

It does not take any sophisticated tools to understand groups of order \(2p\), 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 \(n=6,\,10,\,14\).

For \(n=9\), 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 \({\mathbb Z}_3\times{\mathbb Z}_3\). 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: \({\mathbb Z}_3\times{\mathbb Z}_5\cong{\mathbb Z}_{15}\).

We have just two orders left to analyze: \(n=8\) and \(n=12\). 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 non-abelian groups are the dihedral group (symmetries of a square) and the quaternions.

For \(n=12\) there are \(2\) abelian groups, and \(3\) non-abelian groups. We know two of the non-abelian groups as a dihedral group, and the alternating group on \(4\) symbols (which is also the symmetries of a tetrahedron). The third non-abelian group is an example of a “dicyclic” group, which is an infinite family of groups, each with order divisible by \(4\). The order \(12\) dicyclic group can also be constructed as a “semi-direct 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 \(2^k\), \(k>2\) 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.

SubsectionGroups of Small Order as Permutation Groups

We list here constructions, as permutation groups in Sage, for all of the groups of order less than \(16\).

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, non-cyclic
5 CyclicPermutationGroup(5) Prime order
6 CyclicPermutationGroup(6) Cyclic
6 SymmetricGroup(3) Non-abelian
7 CyclicPermutationGroup(7) Prime order
8 CyclicPermutationGroup(8) Cyclic
8 C2=CyclicPermutationGroup(2)
G=direct_product_permgroups([C2,C4]) Abelian, non-cyclic
8 C2=CyclicPermutationGroup(2)
G=direct_product_permgroups([C2,C2,C2]) Abelian, non-cyclic
8 DihedralGroup(4) Non-abelian
8 QuaternionGroup() Quaternions
9 CyclicPermutationGroup(9) Cyclic
9 C3=CyclicPermutationGroup(3)
G=direct_product_permgroups([C3,C3]) Abelian, non-cyclic
10 CyclicPermutationGroup(10) Cyclic
10 DihedralGroup(5) Non-abelian
11 CyclicPermutationGroup(11) Prime order
12 CyclicPermutationGroup(12) Cyclic
12 C2=CyclicPermutationGroup(2)
G=direct_product_permgroups([C2,C6]) Abelian, non-cyclic
12 DihedralGroup(6) Non-abelian
12 AlternatingGroup(4) Non-abelian
Symmetries of tetrahedron
12 DiCyclicGroup(3) Non-abelian
Semi-direct product \(Z_3\rtimes Z_4\)
13 CyclicPermutationGroup(13) Prime order
14 CyclicPermutationGroup(14) Cyclic
14 DihedralGroup(7) Non-abelian
15 CyclicPermutationGroup(15) Cyclic
Table13.20The Groups of Order \(\mathbf{16}\) or Less in Sage