Equivalence of definitions of group

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This article gives the statement, and possibly proof, of a basic fact in group theory.
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This article gives a proof/explanation of the equivalence of multiple definitions for the term group
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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.


Importance of this equivalence

From the category-theoretical viewpoint

Further information: Groups form a full subcategory of semigroups

The equivalence of these two definitions tells us that the forgetful functor from the category of groups to the category of semigroups is injective: two different groups cannot map to the same semigroup.

Combining this equivalence with the equivalence of definitions of group homomorphism, we obtain that the forgetful functor from groups to semigroups is full, faithful and injective, so the category of groups is a full subcategory of the category of semigroups.

From the universal algebra viewpoint

The universal algebraic definition of groups is better from the universal algebraic viewpoint because it defines groups as an equational variety; hence, we can apply all the standard constructions for varieties like subalgebras, quotients and direct products. Equating this with a more economical textbook definition also has its uses.


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.