Group (mathematics)

In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. For example, the set of integers is a group under the operation of addition. The branch of mathematics which studies groups is called group theory.

Notation for groups

Usually the operation, whatever it really is, is thought of as an analogue of multiplication, and the group operations are therefore written multiplicatively.

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That is:

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• We write "a · b" or even "ab" for a * b and call it the product of a and b;
• We write "1" for the identity element and call it the unit element;
• We write "a⁻¹" for the inverse of a and call it the reciprocal of a.

However, sometimes the group operation is thought of as analogous to addition and written additively:

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• We write "a + b" for a * b and call it the sum of a and b;
• We write "0" for the identity element and call it the zero element;
• We write "−a" for the inverse of a and call it the opposite of a.

Usually, only abelian groups are written additively, although abelian groups may also be written multiplicatively. When being noncommittal, one can use the notation (with "*") and terminology that was introduced in the definition, using the notation a−1 for the inverse of a.

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If S is a subset of G and x an element of G, then, in multiplicative notation, xS is the set of all products {xs : s in S}; similarly the notation Sx = {sx : s in S}; and for two subsets S and T of G, we write ST for {st : s in S, t in T}. In additive notation, we write x + S, S + x, and S + T for the respective sets.

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