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A collection of objects, with a binary operation, satisfying the group axioms:
I: the result of the operation of two members of the group is always a member of the group ('closure');
II: the operation is associative;
III: the operation has an identity element which is a member of the group;
IV: each element has an inverse element which is a member of the group.

A set G equipped with an associative binary operation ° containing an identity element e such that e °x = x = x °e and an inverse element x⁻¹ satisfying x °x⁻¹=e=x⁻¹ °x for each x ∈ G .
A subgroup H ≤ G is a subset closed under the group operations °, e, ( )⁻¹ and so forms a group in its own right. Given H ≤ G , we can form the quotient G/H = { gH | g ∈ G } , which partitions G into subsets of equal size.
Lagrange's theorem states that the order of a subgroup divides the order of a group, and in the finite case, |G|=|H|.|G : H| where |G : H| , the index of G in H is equal to |G/H| . In particular, the order of an element divides the order of the group. Note that the converse to Lagrange's theorem is false; for example, there are no subgroups of the group A₄ of even permutations of order 6, although |A₄|=12 . On the other hand, if p | |G| for prime p then G contains elements of order p .
See also permutation, monoid, normal subgroup, quotient group.