2 Overview of Group Theory Group theory is an area of mathematics that deals with a collection of objects, called a set, along with some operation between them, called a binary operation. A set with a binary operation that together satisfy certain requirements is called a group. As an example of this, let us think about D4, the group of symmetries of a square, or the ways a square can be moved without changing its appearance. D4 has 8 elements. We define y to be a rotation by π 2 and x to be reflection. Hence, our eight elements are . The Elements of D4 Composition of these functions is our binary operation. These together form a group. For a set Gand a binary operation ∗, the group is denoted by (G, ∗). If we want to perform the binary operation on two elements in the set, say a, b ∈ G, we denote that by a ∗ b. Note that a ∗ b must be in Gfor any a and b in G. We often ignore the binary operation when denoting a group, and instead simply write the symbol for the set. In other words, instead of writing (G, ∗) every time the group is mentioned, we will often denote the group as G. Additionally, multiplicative or additive notation is often used for the binary operation, although the operation may not be multiplication or addition. For G to be a group, a few qualifications must be met. A key qualification is that every group must have an identity element, usually denoted e, such that a∗e =e∗a =a for all a in G. Going back to our previous example with the square group, doing nothing with the square is the identity element, because if you do nothing with the square and then move it in some way it is the same as just moving it in some way without purposefully doing nothing first. This qualification is important for this discussion because it will play a role when we define simple groups. Page 28 Riley • An Overview of Monstrous Moonshine
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