Group theory and generalizations Example

Variety of groups

Example

Take the signature with a binary operation $\cdot$ , a unary operation ${}^{-1}$ and a constant $e$ . The variety of groups is the class of algebras $(G, \cdot, {}^{-1}, e)$ of this signature that satisfy the identities $(x \cdot y) \cdot z = x \cdot (y \cdot z), \qquad x \cdot e = e \cdot x = x, \qquad x \cdot x^{-1} = x^{-1} \cdot x = e.$ These algebras are exactly the groups, with the inverse map and the identity element listed as operations. The class is a variety, since it is defined by identities. In this signature, a subalgebra of a group is a subgroup, a homomorphism is a group homomorphism, and a congruence corresponds to a normal subgroup.

Remarks

The choice of signature matters. With the binary operation $\cdot$ alone, the class of groups is not closed under subalgebras. For example, $(\N, +)$ is closed under $+$ inside the group $(\Z, +)$ , but it is not a group. So the class of groups is not a variety in that signature. Subvarieties of the variety of groups include the abelian groups, defined by $x \cdot y = y \cdot x$ , and the groups of exponent $n$ , defined by $x^n = e$ .

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Group concept
Identity concept
Variety of groups example

Tag	`FX8CRCAYXZ`
URI	`m2://FX8CRCAYXZ`
Kind	example
Writing	W0 stub
MSC	20E10, 08B99
Instance of	Variety of algebras
Uses	Group, Identity
Changed	2 November 2026

Variety of groups

Example

Remarks

Reading path

Further reading

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