# Variety of groups

This article describes a way of viewing the collection of groups as a structure in its own right

## Definition

### As a plain variety

The variety of groups is defined as the variety of all algebras with an operator domain comprising a binary operation $*$, a constant operation $e$, and a unary inverse map $()^{-1}$, satisfying three conditions:

• The associativity condition on $*$: $(a * b) * c = a * (b * c)$

• The neutral element condition on $e$ with respect to $*$: $a * e = e * a = a$

• The inverse element condition on the inverse map with respect to $*$: $a * a^{-1} = a^{-1} * a = e$

All these conditions are universally quantified. An algebra of this variety is thus a set equipped with the three operations, such that the above three conditions hold for all choices of elements $a,b,c$ in the set.

The algebras of this variety are precisely the groups.

### As a variety with zero

A variety with zero is a variety of algebras with a particular constant operation distinguished as its zero operation. The variety of groups naturally gets such a structure: choose the zero operation to be the constant map to the neutral element (or the identity element).

Since the variety of groups has only one constant operation, the zero is more or less forced on us, so we shall not distinguish between the variety itself and the variety with zero.

Note that the zero here is particularly nice because it the zero element in any group is a subgroup of it (namely the trivial subgroup). The analogous statement is not true for many other algebraic varieties -- for instance, in the variety of unital rings with the zero being the zero of the ring, zero is not a subobject.

## Properties

A complete listing of universal algebra-theoretic properties satisfied by the variety of groups is available at: