Tour:Equivalence of definitions of group

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This article adapts material from the main article: equivalence of definitions of group

This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)
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PREREQUISITES: The two definitions of group, the statement about equality of left and right neutral element and the statement about equality of left and right inverses

WHAT YOU NEED TO DO:

  • Revisit the two equivalent definitions of group given in the page on group. In one language, we emphasized on the existence of three operations: a binary operation called the group multiplication, a unary operation that sends every element to its inverse, and a 0-ary operation that maps everything to the identity element (also called the neutral element). The other definition just talked of a binary operation, satisfying certain conditions.
  • Read below, and understand thoroughly, why the definitions are equivalent (the two crucial facts used are uniqueness of identity element and uniqueness of inverses)

The definitions that we have to prove as equivalent

Textbook definition (with symbols)

A group is a set G with a binary operation * such that the following hold:

  • For any a,b,c in G, (a * b) * c = a * (b * c). This property is termed associativity.
  • There exists an element e in G such that a * e = e * a = a for all a in G. Such an e is termed a neutral element or identity element for G.
  • For any a in G, there is an element b such that a * b = b * a = e. Such a b is termed an inverse of a and is denoted as a^{-1}.

From the above definition, we can prove that there is only one identity element and every element has a unique inverse.

Universal algebraic definition (with symbols)

A group is a set G equipped with three operations:

  • A binary operation * (infix operator)
  • A 0-ary operation which gives a constant element, denoted as e
  • A unary operation {}^{-1} (superscript operator)

satisfying the following three compatibility conditions:

Notice that in this latter definition, all the compatibility condition are in the form of universally quantified equations. These show that groups form a variety of algebras and the techniques of universal algebra can be applied to them.

The key difference between the definitions

The main difference is that the first definition (textbook definition) only postulates existence of an identity element (neutral element) and inverses, but does not include them as part of the group structure.

The second definition (universal algebra definition) actually specifies a constant to be called the identity element (neutral element), and a unary operation that plays the role of the inverse map. These are therefore part of the group structure in the universal algebra definition.

To show the equivalence, we really need to show that the identity element and inverse map of a group are already uniquely determined by the binary operation.


Facts used

  1. Binary operation on magma determines neutral element, following in turn from equality of left and right neutral element
  2. Two-sided inverse is unique if it exists in monoid, which in turn follows from equality of left and right inverses in monoid

Proof

What we essentially must show

We need to show that if G is a group, the binary operation uniquely determines both the inverse map and the neutral element. From that, it will follow that the textbook definition which asserts existence of a neutral element and of inverses, can be converted to the universal algebraic definition which specifies the neutral element and inverses as part of the group structure.

Uniqueness of neutral element

By Fact (1), it is true that for any magma (set with a binary operation) there can be at most one neutral element (identity element). Thus, in particular, in the case of a group, an identity element does exist, hence it is unique.

Uniqueness of inverse

By Fact (2), in a monoid (set with associative binary operation having neutral element), there can be at most one inverse element to a given element. That is, for any a, there can be at most one b such that a * b = b * a = e.

In other words, the inverse operation in a group is uniquely determined from the multiplication and the identity element. Since we've already shown that the identity element is determined by the multiplication, we have established that the inverse operation is uniquely determined by the group multiplication.


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