"I suspect there is a group theoretic description because it's a vertex-transitive graph."
"By adding group relations, you make them connected."
"When we find a set that is symmetric under a certain kind of actions, we call it a group."
"Once we have the information about the right group, we will use the second equality to deduce the structure of the group in the center."
"A first fundamental point is that it is very natural to interpret the elements of a group as actions, as movements."
"And the collection of all such action is a subgroup of the automorphism group of the left."
"And we know exactly what simple groups make up any finite abelian group, which are all the prime cyclic groups."
"The group of two by two unitary matrices with a determinant equal to plus one is called SU2, the special unitary group of two by two matrices."
"Group Theory celebrates identifying patterns such as the fact that the pentagons and the two-digit pairs we saw are effectively the same because their Cayley tables are the same."
"The SU(2) group has exactly one unique representation for a given matrix size."
"A representation is a function rho that takes every member in a group G and assigns it to an invertible n-by-n matrix."
"SU(2) is defined as the set of 2x2 complex unitary matrices with determinant 1."
"A group has a multiplication, it has an identity element, and it has an inverse for each element."
"So what is a Lie group? Well, a Lie group, as the name suggests, is a group and it's also a manifold."
"The function taking lambda to the exponential of lambda A is a homomorphism of groups from the reals to the group."
"An automorphism of a group G was an isomorphism from that group to itself."
"An abelian group means that R plus S equals S plus R."
"The group of rational points on an elliptic curve is actually a finitely generated abelian group."
"The group structure on the solutions of an elliptic curve shows you that you have a good chance of constructing a lot of points."
"The additive group of real numbers is isomorphic to the multiplicative group of positive real numbers."
"Kaylee's theorem says yes, that if we've got an abstract group, then there's some object it's the symmetries of."
"If any two elements of a group commute, then we say G is commutative or sometimes abelian."
"Any group has to contain an identity element which happens to be its own inverse."
"A group is by definition a set with a product structure."
"Every finite group is secretly isomorphic to some collection of permutations."
"The cyclic group \(C_n\) is generated by a single rotation \(r\), subject to \(r^n\) being the identity rotation."
"The multiplication tables for cyclic groups are cyclical shifts to the left of the row above it."
"The multiplication table must be symmetric across the main diagonal for abelian groups."
"The cyclic group of order \(n\), i.e., \(n\) rotations, is denoted \(C_n\) or sometimes \(Z_n\)."
"Instead of thinking of the smallest normal subgroup for which the quotient is abelian, you can look at the quotient and say what is the largest thing you can get that's abelian."
"There is always a unique homomorphism from the abelianization to that same group such that our original homomorphism can be done by first doing the quotient and then doing this unique homomorphism."
"If we have a group \( G \) and any quotient to a fixed abelian group \( A \), what we could do is we could first quotient out by the commutator to get an abelian quotient."
"Instead of this big collapsing down to an abelian group, you can first collapse minimally by quotienting out the commutator subgroup."
"This homomorphism \( F \) factors through the abelianization."
"The commutator subgroup of \( D_4 \) is the subgroup generated by \( R^2 \)."
"By the correspondence theorem, the abelianization of \( A_4 \) is the quotient of \( A_4 \) with this, so the quotient must be a subgroup of order three."
"If we quotient by anything above this commutator subgroup, we automatically get an abelian quotient."
"Group theory is now a major part of mathematics and how we analyze symmetries."
"Isomorphisms are a very useful and important idea in our study of groups."
"Isomorphism is a map between two groups which is bijective and such that it preserves the multiplication operation."
"Every element in the group has another element called its inverse, whose product with it is the identity element."
"If f of X has degree n, then the Galois group of F is isomorphic to a subgroup of S_n."
"Two groups are isomorphic if there exists a mapping which maps one group onto the other, where the mapping is bijective."
"If you think about it just right, you can represent a group as a collection of symmetries."
"The remarkable thing about mathematics is that suddenly for a while it looks just like things about abstract notions like that of a group."
"The question was, will it go on forever like this, or is there a limit to them? And recently, it has been proved that there are no more, but the list includes all possible finite simple groups."
"A group G acts on a set s if there is a homomorphism from G into the set of permutations of s."
"The order of any subgroup must divide the order of the group."
"If H satisfies any of these five equivalent conditions, we say that H is a normal subgroup."
"A simple group is a group with no non-trivial normal subgroup."
"Every finite abelian group is isomorphic to a direct product of cyclic groups."
"Every finitely generated abelian group is isomorphic to a direct product of cyclic groups."
"Thinking about linear transformations is often a much more convenient way to think about group elements."
"The alternating group on N symbols is going to consist of all the permutations in SN that are even."
"The direct product of groups is a method for making larger groups from smaller groups."
"The direct product of two groups joins them so they can act independently of each other."
"To formalize the notion of symmetry, we use group actions."
"The orbit of X is just going to be the set of all Y in our set X such that X and Y are related."
"The addition composition table has to obey some incredible properties, the properties of an abelian group."
"The action of the absolute Galois group is the most important of them."
"A homomorphism is a function between two groups satisfying the following important property: Fe of a * b equals Fe of a times F of b."
"Any finite subgroup of G is isomorphic either to a cyclic group or a dihedral group."
"The next thing we're going to do is not classify the finite groups of this motion group but are what are called the discrete groups, and that's where we get into the beautiful study of symmetry."
"For a finite group, no matter how large, we can add up those vectors, divide by the number of vectors, that gives us a new vector in the plane, and that's the fixed point."
"Every arrow is an isomorphism in a group."
"Every group G is isomorphic to a group of permutations."
"A group acting on itself by conjugation has orbits that are its conjugacy classes."
"Groups act on cosets of a subgroup by right multiplication."
"Cycle decomposition is a really important and useful concept."
"For all elements of the group, there must exist another element which will compose with it to give the identity."
"The elements of the group represent permutations of some set."
"The composition law represents composing two of the set permutations together."
"The identity element is just representing the identity set permutation, where all elements of the set are mapped onto themselves."
"If you've got any set permutation represented within your group, then you also have the exact inverse map also represented within your group."
"The group must contain an identity element, which will compose with any other element to give that other element back again."
"Infinite cyclic subgroups will end up being isomorphic to infinite cyclic groups."
"The finite cyclic groups are a whole class of groups."
"For fee to be a homomorphism, it must satisfy this equation: \( \phi(a \cdot b) = \phi(a) \cdot \phi(b) \) for all \( a \) and \( b \) in the group."
"There always exists a subgroup capital H which is a subgroup of capital G such that the order of capital H is equal to P I to the power of alpha I for any I that you like between 1 and R."
"The first Sylow theorem is probably the most important, and it's also the easiest to understand."
"Subgroups of order P to the power of alpha, where alpha is the highest power that P appears in the prime factorization of the order of the group, are now called Sylow P subgroups."
"It would be good to understand the notion of a normal subgroup."
"A lot of algebraic topology is about a way of using group theory to describe the fact that certain topological spaces have holes."
"Grand unified theories... are based on group Theory."
"The group Theory extends the SU(3) cross SU(2) cross U(1) of the standard model."
"SU(2) classifies the spin representations of particles."
"The dihedral group of order four is the Klein four group."
"A group action on a set is a map that takes an element in a group and an element in the set to another element in a set."
"G acts transitively on S, you can get from any one left coset of K to any other left coset of K by multiplying by an element in G."
"It is often helpful to think of the elements of a group as representing set permutations of some set."
"We're trying to develop methods for constructing new groups from old groups."
"The quotient group Q mod Z is infinite, but every element in this group has finite order."
"If we know how to find that automorphism group, then we're going to be able to get some deep information about how the roots of that polynomial are related one to another."
"If these four properties satisfy for a set along with this operator, then it is called a group."
"This is a deep deep insight into the nature of the composition table on groups."
"Finite simple groups is one of the great tales of 20th century mathematics."
"If we have two arbitrary cosets, \( \bar{a} \) and \( \bar{b} \), and we want to compose them together... the answer to \( \bar{a} \) composed with \( \bar{b} \) is going to be equal to the coset that contains \( a \) composed with \( b \)."
"Providing the subgroup we're using is a normal subgroup, you can pick any representative from the coset \( \bar{a} \), any representative from the coset \( \bar{b} \), compose the two of them together, and you'll always get an answer that's in the same coset."
"Whenever you compose two cosets together, you will end up with another coset."
"The composition of the cosets in our quotient group... does obey associativity."
"The identity element in our quotient group is going to be the normal subgroup itself."
"For any coset in the quotient group, you have another coset which will compose with it either way around to give the identity element."
"The center of an of a group of prime power order is non-trivial."
"If the order of G is equal to P, G is cyclic."
"In the case of order 15, there's only one group and it's cyclic."
"The number L of such subgroups satisfies L divides M and L is congruent to 1 mod P."
"The left coset of a subgroup H under an element a is equal to the set of all things of the form a composed with h."
"You can never have a subgroup which doesn't have an order which divides the order of the group."
"An automorphism of our group G is an isomorphism Pi carrying our group back to itself."
"It allows us to solve polynomial equations by exploiting symmetries coming from group theory."
"This will form a group, not only a group, it is a cyclic group."
"Understanding conjugacy in a group is one of the most important things to be able to do."
"Two permutations which are conjugate in S_n have the same cycle shape."
"An automorphism is going to be a relabeling of the elements of the group in a way that preserves multiplication."
"We have a theorem that states: In G is isomorphic to the quotient group G by the center of G."
"For a concrete group, there will be additional theorems that are true about that group which are not true for the universal algebraic case."
"If you have two groups that are isomorphic to one another, their lattice diagrams are going to be identical."
"A sub n is simple when n is greater than or equal to 5."
"There are no normal subgroups in A sub n except for the identity and A sub n itself."
"If two groups are isomorphic to each other, then they're basically the same under the hood, share the same properties, and work exactly the same way."
"The mapping from the Li algebra to the Li group is done by the exponentiation."
"The order of a group G is the number of elements in G."
"We'll start with a very simple but beautiful concept: groups."