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:

Category:Property satisfactions for the variety of groups

Good congruence-related properties

• Variety of groups is ideal-determined: There is a correspondence between congruences and ideals, via the map which sends a congruence to its kernel (the congruence class of zero). Ideal here is a generic notion that can be described for any variety with zero. In the particular case of groups, the ideals are simply the normal subgroups. Analogous results are true for many varieties of algebras which have groups as a reduct; for instance, unital rings, Abelian groups.
• Ideals are subalgebras in the variety of groups: Every ideal in the variety of groups is a subalgebra, and thus the kernel of any congruence is a subalgebra. In the plain language of group theory, this is the fact that every normal subgroup is a subgroup. Note that the converse is not true for groups, though it is true in the variety of Abelian groups (a converse statement in general requires some kind of Abelianness).
• Characteristic subalgebras are ideals in the variety of groups: This is the universal algebraist's way of saying that characteristic subgroups are normal. Combining with the fact that groups are ideal-determined, this yields that given any characteristic subalgebra, we can take a quotient algebra by the induced congruence.
• Variety of groups is congruence-permutable: This translates to the fact that a product of normal subgroups is again a subgroup and moreover, is normal.
• Variety of groups is congruence-uniform: This translates to the fact that all the cosets of a normal subgroup have equal size.