# Definition:Group Axioms

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## Contents

## Definition

A group is an algebraic structure $\struct {G, \circ}$ which satisfies the following four conditions:

\((G \, 0)\) | $:$ | Closure | \(\displaystyle \forall a, b \in G:\) | \(\displaystyle a \circ b \in G \) | ||||

\((G \, 1)\) | $:$ | Associativity | \(\displaystyle \forall a, b, c \in G:\) | \(\displaystyle a \circ \paren {b \circ c} = \paren {a \circ b} \circ c \) | ||||

\((G \, 2)\) | $:$ | Identity | \(\displaystyle \exists e \in G: \forall a \in G:\) | \(\displaystyle e \circ a = a = a \circ e \) | ||||

\((G \, 3)\) | $:$ | Inverse | \(\displaystyle \forall a \in G: \exists b \in G:\) | \(\displaystyle a \circ b = e = b \circ a \) |

These four stipulations are called the **group axioms**.

## Also known as

The **group axioms** are also known as the **group postulates**, but the latter term is less indicative of the nature of these statements.

The numbering of the axioms themselves is to a certain extent arbitrary. For example, some sources do not include $G \, 0$ on the grounds that it is taken for granted that $\circ$ is closed in $G$. However, in the treatment of more abstract aspects of group theory it is recommended that this axiom be taken into account.

## Also see

## Sources

- 1964: Walter Ledermann:
*Introduction to the Theory of Finite Groups*(5th ed.) ... (previous) ... (next): $\S 2$: The Axioms of Group Theory - 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 4.4$. Gruppoids, semigroups and groups - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: The Definition of Group Structure: $\S 27$ - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 5$: Groups $\text{I}$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 33$. The definition of a group - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms - 1996: John F. Humphreys:
*A Course in Group Theory*... (next): Chapter $1$: Definitions and Examples: Definition $1.1$