From: 3blue1brown
The Monster Group is a number approximately 8x10^53, a value comparable to the number of atoms in the planet Jupiter [00:00:16]. This number is considered peculiar and reflective of something fundamental in mathematics [00:00:35]. It represents the size of the Monster Group [00:00:45].
Introduction to Groups and Symmetry
Group theory is a field focused on codifying the idea of symmetry [00:00:52]. A face is symmetric if it can be reflected about a line and remain unchanged [00:00:56]. A snowflake is symmetric in multiple ways, allowing rotations (e.g., 60 or 120 degrees) and flips along various axes, all of which leave its appearance unchanged [00:01:08]. A collection of all such actions that leave an object looking the same is called a group [00:01:20]. Mathematicians consider the action of doing nothing to also be part of a group [00:01:41]. For example, the group of symmetries of a snowflake, including the do-nothing action, contains 12 distinct actions and is called D6 [00:01:46]. The simple group of symmetries for a face, having only two elements, is named C2 [00:01:56].
The actions describing symmetries always preserve an implicit structure [00:02:12].
- A cube has 24 rotations that preserve its appearance, forming a group [00:02:17].
- If reflections are allowed, not preserving orientation, the group becomes larger with 48 actions [00:02:27].
- If components (like faces) are less rigidly attached and can be shuffled, the group of “symmetries” becomes much larger and more complicated, reflecting a looser sense of preserved structure [00:03:00].
Permutation Groups
The loosest sense of structure involves a collection of points where any shuffling, or permutation, is considered a symmetry [00:03:08]. These “permutation groups” are unconstrained by underlying properties and can become very large [00:03:18].
- For six objects, there are 6! (720) possible permutations [00:03:41].
- If these points are given structure (e.g., corners of a hexagon with distance preservation), the number of symmetries reduces to 12 (like the snowflake symmetries) [00:03:45].
- For 12 points, the number of permutations is approximately 479 million [00:03:59].
- For 101 objects, there are 101! (around 9x10^159) different actions [00:04:17]. These permutation groups are known as S-sub-n and are fundamental in group theory, encompassing all other groups in a certain sense [00:04:37].
Applications of Group Theory
Early applications of group theory revealed insights into solutions for polynomial equations [00:04:53].
- There are formulas for quadratic, cubic, and quartic equations [00:05:04].
- However, mathematicians struggled to find a formula for degree 5 polynomials [00:05:26].
- Group theory shows that no quintic formula (using only arithmetic operations and radicals) can exist [00:05:34]. This impossibility relates to the inner structure of the permutation group S5 [00:06:04].
A theme in math is that the nature of symmetry can reveal non-obvious facts about other objects [00:06:13]. In physics, Noether’s theorem states that every conservation law corresponds to some kind of symmetry or group [00:06:27]. This means fundamental laws like conservation of momentum and energy correspond to groups [00:06:35].
Abstracting Groups
While groups can be understood as collections of actions, mathematicians define them more abstractly [00:07:10]. Similar to how the number 3 is an abstract concept rather than a specific triplet of things, group elements are abstract entities [00:07:18]. What defines a group is how its elements combine [00:08:04]. Combining actions means applying one after another (e.g., flipping a snowflake then rotating it results in an equivalent single action) [00:08:10]. This defines a “multiplication” operation [00:08:31]. A multiplication table captures the inner structure of a group, abstracted away from the specific object it acts on (like a square or polynomial roots) [00:09:03].
Formally, a group is a set with a binary operation (multiplication) that satisfies four axioms [00:09:59]. These axioms derive from properties that are inherently true when thinking about composing actions [00:10:19].
Isomorphism
Abstraction in group theory is desirable because different situations can lead to the same underlying group [00:10:42]. For instance, the symmetries of a cube and the permutation group of four objects (S4) are fundamentally the same [00:10:46]. This means their multiplication tables are identical, and anything true for one is true for the other [00:11:07]. This relationship is called an isomorphism, which is a one-to-one mapping between group elements that preserves composition (multiplication) [00:12:18]. An important example is how the rotations of a cube permute its four diagonals [00:12:34]. Seeing the same group in different contexts reveals unexpected connections between distinct objects [00:13:17].
The Classification of Finite Simple Groups
A natural question in group theory is: “What are all the groups?” [00:13:34] More precisely, “What are all the groups up to isomorphism?” meaning groups are considered the same if an isomorphism exists between them [00:13:39]. This is asking about all the ways something can be symmetric [00:13:53].
The question is exceedingly hard [00:14:05]. Groups are divided into infinite groups (like symmetries of a line or circle) and finite groups [00:14:10]. Focusing on finite groups, they can be broken down into “smaller groups” in a way analogous to prime factorization for numbers or atoms for molecules [00:14:25]. Groups that cannot be broken down any further are called simple groups [00:14:38]. For instance, the proof that no quintic formula exists involves showing that the permutation group on 5 elements (S5) decomposes into a type of simple group (cyclic groups of prime order) that polynomial solutions from radicals would not allow [00:14:48]. Understanding these “atomic” simple groups is crucial outside of group theory [00:15:37].
The task of categorizing all finite groups has two steps:
- Find all the simple groups [00:15:45].
- Find all the ways to combine them [00:15:50].
Mathematicians have successfully found and proven that they have found all finite simple groups [00:16:00]. This monumental achievement took decades, tens of thousands of pages, hundreds of minds, and computer assistance, culminating by 2004 [00:16:12].
The result is considered “absurd” [00:16:33]:
- There are 18 distinct infinite families of simple groups [00:16:36].
- One family includes cyclic groups with prime order (symmetries of a regular polygon with a prime number of sides, without flips) [00:17:08].
- Another is similar to permutation groups (S-sub-n), simple if acting on 5 or more elements, which relates to why quintic polynomials lack radical solutions [00:17:21].
- There are also 26 sporadic groups that do not fit into these patterns [00:16:43]. This “patched together” fundamental structure is seen as bizarre [00:16:54].
The Monster Group
The largest of these sporadic groups is named the Monster Group [00:17:58], with its size being 8x10^53 [00:18:02]. The second largest is the Baby Monster Group [00:18:06]. Nineteen of the sporadic groups, including the Baby Monster, are considered “children of the monster” and are called the “happy family” by Robert Gries [00:18:11]. The remaining six are called “pariahs” [00:18:20].
The Monster Group is not just big, but its abrupt presence as a fundamental building block is strange [00:18:32]. The Monster acts on an object in 196,883 dimensions [00:18:59]. Describing one element of this group requires about 4 GB of data, despite other much larger groups having smaller computational descriptions [00:19:17].
Why the Sporadic Groups? (Monstrous Moonshine)
The existence of sporadic groups, particularly the Monster, remains largely mysterious [00:19:42]. In the 1970s, John McKay noticed a number very similar to 196,883 (one greater) appearing in the series expansion of a fundamental function related to modular forms and elliptic functions [00:20:00]. This observation, initially deemed “moonshine” by John Conway due to its seemingly crazy nature, led to the “monstrous moonshine conjecture” after more numerical coincidences [00:20:25]. Richard Borcherds proved this conjecture in 1992, establishing a connection between disparate areas of mathematics [00:20:40]. This proof contributed to him winning the Fields Medal six years later [00:20:48]. Monstrous Moonshine also has a connection to string theory [00:20:53].
The Monster Group serves as a reminder that fundamental objects in mathematics are not necessarily simple, and the universe’s answers don’t prioritize human understanding [00:21:09].