Abstract Algebra/Group Theory/Group/Definition of a Group/Definition of Inverse

Inverse:
1. if c is in G, c^-1 is in G.
2. c*c^-1 = c^-1*c = e_G

Definition of Inverse

Let G be a [[../|group]] with operation $\ast$

$\mathrm{\forall }g\in G:\mathrm{\exists }{g}^{-1}\in G:g\ast g^{-1}=g^{-1}\ast g=e_G$

Usages

1. Template:AnchorIf g is in G, g has an inverse g⁻¹ in G
2. Template:Anchorb is the inverse of g on group G if

   b is in G, and

   b $\ast$ g = g $\ast$ b = e_G.

   e_G here again means the Identity of group G.

3. Template:AnchorIf b is the inverse of g on group G, then

   b is in G, and

   b $\ast$ g = g $\ast$ b = e_G.

Notice

1. G has to be a [[../|group]]