Chapter 16
Introductory Group Theory
A group comprises mathematical elements that satisfy four simple definitions. A
bunch of groups exists under these simple definitions. This makes the group theory a
discriminating field of mathematics. To get familiar with various concepts of groups,
we first show several tangible examples. Group elements can be numbers (both real
and complex) and matrices. More abstract mathematical elements can be included as
well. Examples include transformation, operation, etc. as already studied in previous
parts. Once those mathematical elements form a group, they share several common
notions such as classes, subgroups, and direct-product groups. In this context,
readers are encouraged to conceive different kinds of groups close to their heart.
Mapping is an important concept as in the case of vector spaces. In particular,
isomorphism and homomorphism frequently appear in the group theory. These
concepts are closely related to the representation theory that is an important pillar
of the group theory.
16.1 Definition of Groups
In contrast to a broad range of applications, the definition of the group is simple. Let
ℊ be a set of elements g ν , where ν is an index either countable (e.g., integers) or
uncountable (e.g., real numbers) and the number of elements may be finite or
infinite. We denote this by ℊ ¼ {g ν }. If a group is a finite group, we express it as
ℊ ¼ g 1 , g 2 , Á Á Á, g n
f
g ,
ð16:1Þ
where n is said to be an order of the group.
Definition of the group comprises the following four axioms with respect to a
well-defined “multiplication” rule between any pair of elements. The multiplication
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_16
621
Introductory Group Theory
A group comprises mathematical elements that satisfy four simple definitions. A
bunch of groups exists under these simple definitions. This makes the group theory a
discriminating field of mathematics. To get familiar with various concepts of groups,
we first show several tangible examples. Group elements can be numbers (both real
and complex) and matrices. More abstract mathematical elements can be included as
well. Examples include transformation, operation, etc. as already studied in previous
parts. Once those mathematical elements form a group, they share several common
notions such as classes, subgroups, and direct-product groups. In this context,
readers are encouraged to conceive different kinds of groups close to their heart.
Mapping is an important concept as in the case of vector spaces. In particular,
isomorphism and homomorphism frequently appear in the group theory. These
concepts are closely related to the representation theory that is an important pillar
of the group theory.
16.1 Definition of Groups
In contrast to a broad range of applications, the definition of the group is simple. Let
ℊ be a set of elements g ν , where ν is an index either countable (e.g., integers) or
uncountable (e.g., real numbers) and the number of elements may be finite or
infinite. We denote this by ℊ ¼ {g ν }. If a group is a finite group, we express it as
ℊ ¼ g 1 , g 2 , Á Á Á, g n
f
g ,
ð16:1Þ
where n is said to be an order of the group.
Definition of the group comprises the following four axioms with respect to a
well-defined “multiplication” rule between any pair of elements. The multiplication
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_16
621