Definition:Group
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Definition
A group is a semigroup with an identity (i.e. a monoid) in which every element has an inverse.
Group Axioms
The properties that define a group are sufficiently important that they are often separated from their use in defining semigroups, monoids, etc. and given recognition in their own right.
A group is an algebraic structure $\left({G, \circ}\right)$ which satisfies the following four conditions:
| \((G0):\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \forall a, b \in G\) | \(:\) | \(\displaystyle a \circ b \in G\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Closure Axiom | |
| \((G1):\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \forall a, b, c \in G\) | \(:\) | \(\displaystyle a \circ \left({b \circ c}\right) = \left({a \circ b}\right) \circ c\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Associativity Axiom | |
| \((G2):\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \exists e \in G: \forall a \in G\) | \(:\) | \(\displaystyle e \circ a = a = a \circ e\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Identity Axiom | The element $e$ is called the identity of $G$. |
| \((G3):\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \forall a \in G: \exists b \in G\) | \(:\) | \(\displaystyle a \circ b = e = b \circ a\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Inverse Axiom | The element $b$ is called the inverse of $a$ and is usually denoted $a^{-1}$. |
These four stipulations are called the group axioms.
Group Product
The notation $\left({G, \cdot}\right)$ is used to represent a group whose underlying set is $G$ and whose operation is $\cdot$.
The operation $\cdot$ is referred to as the group product or just product.
Multiplicative Notation
When discussing a general group with a general group product, it is customary to dispense with a symbol for this operation and merely concatenate the elements to indicate the product.
That is, we invoke the multiplicative notation and write $a b \in G$ instead of $a \cdot b \in \left({G, \cdot}\right)$, as this can make the notation more compact and the arguments easier to follow.
Compare with additive notation.
Also denoted as
Some sources use the notation $\left \langle G, \circ \right \rangle$ for $\left({G, \circ}\right)$.
Historical Note
The concept of the group as an abstract structure was introduced by Arthur Cayley in his 1854 paper On the theory of groups.
The first one to formulate a set of axioms to define the structure of a group was Leopold Kronecker in 1870.
Also see
Sources
- Walter Ledermann: Introduction to the Theory of Finite Groups (1949): $\S 2$: Definition $1$
- Iain T. Adamson: Introduction to Field Theory (1964)... (previous)... (next): $\S 1.1$
- J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 4.4$
- Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 7$
- Richard A. Dean: Elements of Abstract Algebra (1966): $\S 1.2$
- George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Chapter $\text{II}$
- B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970): $\S 1.1$: Definitions $1.1 \ \text{(b)}$
- Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 26, \ \S 27$
- Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 33$
- John F. Humphreys: A Course in Group Theory (1996): $\S 1$: Definition $1.1$
