Understanding the Notation of U(p,q) in Group Theory | Wybourne Book Study

In summary, group theory is a branch of mathematics that studies the properties of groups and has applications in various fields such as physics, chemistry, and computer science. The basic concepts in group theory include groups, subgroups, cosets, cyclic groups, normal subgroups, and group homomorphisms. It is used in science to understand and describe symmetries and patterns in physical systems. Real-life examples of groups include integers, matrices, and symmetries of polygons. Important theorems in group theory include Lagrange's theorem, the first isomorphism theorem, Sylow theorems, the classification of finite simple groups, and Cayley's theorem.
  • #1
somy
137
0
Hi all.
I'm studying Wybourne book on group theory. I didn't understand this expression:
U(p,q)
I know what U(p+q,C) and U(n) means, but I'm unfamiliar with the notation of the above statement.
Thanks in advance.
Somy:smile:
 
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  • #2
I think U(p,q) denotes the group of complex *unitary* transformations A with respect to a metric g with p plusses and q minusses, i.e
A* g A = g

where A* is the standard hermitian conjugate.
 
  • #3
I think so.
Thanks dear Careful!
 

FAQ: Understanding the Notation of U(p,q) in Group Theory | Wybourne Book Study

What is group theory?

Group theory is a branch of mathematics that studies the properties of groups, which are mathematical structures composed of elements and operations that satisfy certain axioms. It has applications in various fields such as physics, chemistry, and computer science.

What are the basic concepts in group theory?

The basic concepts in group theory include groups, subgroups, cosets, cyclic groups, normal subgroups, and group homomorphisms. These concepts help define and understand the structure and behavior of groups.

How is group theory used in science?

Group theory has many applications in science, such as in quantum mechanics, crystallography, and symmetry analysis. It provides a useful framework for understanding and describing the symmetries and patterns in physical systems.

What are some real-life examples of groups?

Some real-life examples of groups include the set of integers under addition, the set of 2x2 matrices with nonzero determinant under multiplication, and the set of symmetries of a regular polygon under composition. Groups can also be used to describe the structure and behavior of molecules in chemistry.

What are some important theorems in group theory?

Some important theorems in group theory include Lagrange's theorem, which states that the order of a subgroup must divide the order of the original group, and the first isomorphism theorem, which states that any group homomorphism induces an isomorphism between the group and its image. Other important theorems include the Sylow theorems, the classification of finite simple groups, and the Cayley's theorem.

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