Given a subset G0 of G(such that (G, ∗) is a group), we say that G0 is a subgroup of Gif G0 is a group under ∗. If G0 is a subgroup of Gwe denote it by G0 ≤G. An important concept for the discussion that will follow is the idea of a simple group. A simple group is a group that has no proper, nontrivial, normal subgroups [4]. By a proper subgroup, we mean a subgroup whose set is not the set of the whole group (in finite groups the set of a proper subgroup is strictly smaller in cardinality than the set of the original group). By nontrivial, we mean that the subgroup must contain more elements than just the identity element. Note that simple groups themselves must be nontrivial. In other words, the group whose set is just the identity element is not a simple group. Groups are classified as finite or infinite depending on the size of the set. A subgroup being normal is a technical condition. We are interested in simple groups since they are the “building blocks” of larger groups. These are the groups that cannot be “factored” down into smaller groups. There are two broad categories of simple groups. They are 18 infinite families of groups and 26 sporadic groups [7]. In this paper, we will be focusing on the largest sporadic group, the Monster(denoted M). 3 The Monster Group In 1973, Fischer and Griess independently conjectured that the Monster group existed. In 1980, Griess constructed it by hand as the automorphism group of a 196884-dimensional commutative non-associative algebra [7]. This algebra is known as the Griess algebra. An automorphism is an isomorphism from a group to itself, and an isomorphism is a bijective function that has the homomorphism property. A homomorphismis a function, say ρ, such that ρ(ab) =ρ(a)ρ(b). As mentioned previously, the Monster is the largest of the sporadic groups. The order of the Monster (the number of elements in the Monster considered as a set) is |M| = 246 · 320 · 59 · 76 · 112 · 133 · 17· 19· 23· 29· 31· 41· 47· 59· 71 = 808, 017, 424, 794, 512, 875, 886, 459, 904, 961, 710, 757, 005, 754, 368, 000, 000, 000. The Monster is generated by involutions, which are elements of order 2 [7]. The order of an element of a group refers to the smallest possible power of the element that is the identity element. The order of any element in a group divides the order of the group. Channels • 2022 • Volume 6 • Number 2 Page 29
RkJQdWJsaXNoZXIy MTM4ODY=