Today, we are going to talk about the Lie groups and , and their respective Lie algebras, generally denoted by and by the physics community. In addition to this, we will see some properties of the orthogonal groups in euclidena … Continue reading

# Category Archives: Mathematics

The converse of the first Lie theorem is also generally true. Theorem. Second Lie Theorem. Given a set of hermitian matrices or operators , closed under commutation with the group multiplication, then these operators define and specify a Lie group … Continue reading

Definition (36). An infinite group is a group where the order/number of elements is not finite. We distinguish two main types of groups (but there are more classes out there…): 1) Discrete groups: their elements are a numerable set. Invariance … Continue reading

Schur’s lemmas are some elementary but very useful results in group theory/representation theory. They can be also used in the theory of Lie algebras so we are going to review these results in this post (for completion). FIRST SCHUR’S LEMMA. … Continue reading

Representation theory is the part of Group Theory which is used in the main applications. Matrices acting on the members of a vector space are assigned to every element of a group. The connections between particle physics and representation theory … Continue reading

We are going to be more explicit and to work out some simple examples/exercises about elementary finite and infinite groups in this post. Example 1. Let us define the finite group of three elements as where , and such as … Continue reading

Other important concepts and definitions in group theory! Definition (22). Normal or invariant group. Let be a subgroup of other group G. We say that is a normal or invariant subgroup of G if the following condition holds: Proposition. Let … Continue reading

Today we are going to speak about two broad and main topics: cyclic groups and some general features of finite groups (a few additional properties and theorems). A cyclic group is, informally speaking, a group that can be generated by … Continue reading

Today we are going to study some interesting aspects of group theory. Definition (14). Subgroup. Given a group and a nonempty subset of G, then H is said to be a subgroup of if and only if Check: If is … Continue reading

Basic definitions of group theory: that is the topic today! We need some background previous to the “group axioms”. Definition (1). Set is a collection of objects with some properties. Objects in the set are called “elements” or “members” of … Continue reading