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Definition of Group: A group is an ordered pair $(G, \circ)$, where:
- $G$ is a set,
- $\circ$ is a binary operation on $G$,
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.
The following statements and definitions logically rely on the material of this page: Def/Abelian, Def/Center (group), Def/Conjugate, Def/Coset, Def/Cyclic, Def/Fixed point, Def/Group homomorphism, Def/Kernel (group), Def/Normal subgroup, Def/Order of a group element, Def/P-group, Def/Ring, Def/Sylow subgroup, Def/Unital, Def/Vector space, State/Cancellation in groups, State/Existence of Sylow subgroups, State/First isomorphism theorem for groups, State/Groups of prime order are cyclic, State/Groups of prime squared order are abelian, State/Homomorphic images of subgroups are subgroups, State/Inversion is an antiautomorphism, State/Kernel criterion for injectivity of group homomorphisms, State/Lagranges theorem, State/Uniqueness of identity element in a monoid, State/Uniqueness of inverse elements in a monoid, and Struct/The group with two elements.
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