Advanced Mathematics

6.4Subgroups: Groups Inside a Group

Purpose of Section

To introduce the concept of a subgroup and find the subgroups of various cyclic groups. We also introduce the idea of cosets of a subset and the concept of a quotient group.

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.

Figure 6.47 Symmetries of an equilateral triangle.

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.

Figure 6.48 Subgroup of rotations of symmetries of an equilateral triangle.

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

Definition

Let {G, *} be a group with operation *. If a subset H ⊆ G itself forms a group with the same operation *, then H is called a subgroup of G.

6.4.1.1 At Least Two Subgroups

Although all groups have two subgroups, ...