6.4.1 Introduction

Recall the six symmetries of an equilateral triangle: the identity map, three flips about the midlines, and two (counterclockwise) rotations of 120° and 240° illustrated in Figure 6.47.

Although we have seen that these symmetries, along with the operation of compositions forms an algebraic group, what is also true is that the group is made up of several smaller groups, In the given case of the six symmetries of an equilateral triangle, the subset of three rotational symmetries {e, R₁₂₀, R₂₄₀} whose multiplication table is shown in Figure 6.48, can easily be verified to form a group.

The above discussion motivates the following definition of “groups within groups,” or subgroups.

6.4.1.1 At Least Two Subgroups

Although all groups have two subgroups, the group itself and the trivial group consist of only the identity {e}, and we are mainly interested in the other subgroups, called proper subgroups, although we often refer to them simply as subgroups.

We let the reader verify each of these subgroups satisfy the required conditions to be a group.

6.4.2 Subgroups of the Klein Four‐Group

Recall that the group of symmetries of a rectangle forms the Klein four‐group with elements

Figure 6.50 shows the Cayley table of these symmetries and three subgroups of order 2.

Note that the order of a subgroup divides the order of the group. This fundamental property of subgroups is called Lagrange's Theorem after the French/Italian mathematician Joseph‐Louis Lagrange (1736–1813).

6.4.3 Test of Subgroups

Although a subset H of a group G is a group only if it satisfies the requirements for a group, it is only necessary to verify that the group operation * is closed in H and that every element of H has an inverse in H. There is no need to show the associative property or the existence of an identity since the identity in G is also an identity in H. This result is summarized in the following theorem.

The subgroups of the symmetry group of a square form a partially ordered set, ordered by set inclusion, illustrated in Figure 6.52. The octic group D₄ is itself a subgroup of the group S₄ of 24 permutations of four elements.

6.4.4 Subgroups of Cyclic Groups

We have seen that the cyclic group Z_n is generated by an element in the group. That is, there exists a g ∈ Z_n such that

To find the subgroups of Z_n, we begin with an arbitrary element h ∈ Z_n and compute the set 〈h〉 generated by h. The elements generated by h may or may not be all of Z_n, but they will always create a subgroup of Z_n. We then pick a second member h^′ ∈ Z_n which is not in the first generated set 〈h〉 and compute 〈h^′〉. This will yield another subset of Z_n. Continuing this process will eventually yield all subgroups of Z_n.

Let us apply this technique to find all the subsets of the cyclic group

where the group operation is addition modulo 12. Starting with 1, we generate powers of g = 1, remembering that powers of one in this group are really adding 1. Hence, we have

which generates the entire group ℤ₁₂. We now select the element g = 2, which generates the subgroup

Figure 6.53 shows the subgroups generated by g = 1, 2, 3, 4. Do you see why 〈5〉 = ℤ₁₂ and 〈6〉 = {0, 6}.

You might notice that the numbers 1, 5, 7, 11 are relative prime to 12 to generate the entire group ℤ₁₂. Do you also see a pattern between the orders of those subgroups generated by numbers not relatively prime with 12? That is, the orders of the subgroups 〈2〉, 〈3〉, 〈4〉, 〈6〉, 〈8〉, 〈9〉, 〈10〉?

Table 6.5 shows the subgroups generated by each element of the group and the order of the subgroup generated by the generator.

6.4.5 Cosets and the Quotient Group

If G is a group with group operation “+” and subgroup H, then for any g ∈ G the “g‐translation” of H

is called the left coset³ of H with respect to g. Similarly, the right cosets are defined by

If the left and right coset coincide,⁴ as they do for commutative groups or for certain subgroups called normal subgroups, we simply refer to them as cosets without specifying left or right. You can think of the cosets of a subgroup as what you get when you “shift” the subgroup to the left or right with members g ∈ G, that is “translated” subgroups. The subgroup H itself is a coset when the “shifting factor” g ∈ G is the identity element in G. By selecting various g ∈ G, one can create a partition of the group G into equivalence classes of cosets. This is useful since subdividing a group into disjoint parts allows one to identify parts of the group that are similar from some point of view.

For example, the (commutative) group G = ℤ₆ of integers mod 6 has a subgroup H = {0, 3} of two elements 0 and 3. Translating the subgroup with the six elements g ∈ ℤ₆, we arrive at three cosets of the form

which are

Note that the subgroup H = {0, 3} itself is a coset when g = 0 ∈ ℤ₆.

The question now arises, is it possible to do “arithmetic” with the cosets, and the answer if yes, when the left and right cosets coincide. Without verification, it is not difficult to show that the binary operation

acting on the cosets forms a group. This group is called the quotient (or factor) group of G modulo H, and are written

pronounced G mod H.

The following example illustrates how an infinite (commutative) group can be “factored” into just three equivalence classes, each having a common property.