Question

what is group in terms of mathematics??​

Answer 1

In mathematics, a group is a set equipped with a binary operation that combines any two elements to form a third element in such a way that three conditions called group axioms are satisfied, namely associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, and help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study.

Answer 2

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A group is a non-empty set GG with one binary operation that satisfies the following axioms (the operation being written as multiplication):

1) the operation is associative, i.e. (ab)c=a(bc)(ab)c=a(bc) for any aa, bb and cc in GG;

2) the operation admits a unit, i.e. GG has an element ee, known as the unit element, such that ae=ea=aae=ea=a for any aa in GG;

3) the operation admits inverse elements, i.e. for any aa in GG there exists an element xx in GG, said to be inverse to aa, such that ax=xa=eax=xa=e.

The system of axioms 1)–3) is sometimes replaced by an equivalent system of two axioms: 1); and 4) the operation admits left and right quotients, i.e. for any two elements aa, bb in GG there exist elements xx, yy in GG, the left quotient and the right quotient of division of bb by aa, such that ax=bax=b, and ya=bya=b.

It follows from this definition that the unit element in any group is unique, that the element inverse to any given element in the group is unique and that for any elements aa, bb of GG both fractions obtained by dividing aa by bb are unique.