# Homotopy Type Theory group > history (changes)

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## Definition

### As a monoid

A group consists of

• A type $G$,
• A basepoint $e:G$
• A binary operation $\mu : G \to G \to G$
• A unary operation $\iota: G \to G$
• A contractible left unit identity
$c_\lambda:\prod_{(a:G)} isContr(\mu(e,a)=a)$
• A contractible right unit identity
$c_\rho:\prod_{(a:G)} isContr(\mu(a,e)=a)$
• A contractible associative identity
$c_\alpha:\prod_{(a:G)} \prod_{(b:G)} \prod_{(c:G)} isContr(\mu(\mu(a, b),c)=\mu(a,\mu(b,c)))$
• A contractible left inverse identity
$c_l:\prod_{(a:G)} isContr(\mu(\iota(a), a)=e)$
• A contractible right inverse identity
$c_r:\prod_{(a:G)} isContr(\mu(a,\iota(a))=e)$
• A 0-truncator
$\tau_0: \prod_{(a:G)} \prod_{(b:G)} isProp(a=b)$

## Examples

• The integers are a group.

• Given a set $A$, the type of automorphisms $Aut(A)$ has the structure of a group, with basepoint $id_A$, binary operation function composition, and unary operation inverse automorphism ${(-)}^{-1}$.