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# Def/Group

Definition of Group: A group is an ordered pair $(G, \circ)$, where:

and the following axioms hold:

• $\circ$ is associative.
• There exists an element (called the identity element) $e \in G$, such that $\forall g \in G, eg = ge = g$.
• For every element $g \in G$, there exists an element (called the inverse of $g$) $h \in G$, such that $gh = hg = e$, where $e$ is the identity element in the previous axiom.

The usage of the phrase " the identity element" is justified by the fact that there is a unique identity element in a group.

The usage of the phrase " the inverse of $g$", in the final axiom, is justified by the fact that there is a unique multiplicative inverse of any element of a group.

## Logical Connections

This definition logically relies on the following definitions and statements: Def/Binary operation, Def/Associative, Def/Composite function

To visualize the logical connections between this definition and other items of mathematical knowledge, you can visit any of the following clusters, and click the "Visualize" tab: Clust/Basic group theory, Clust/Abstract algebraic structures