Question

define weak groups with example​

Answer 1

Explanation:

Usually, we see a group as the structure ⟨G,⋅⟩, where G≠∅ and the binary operation ⋅ is associative and

∃e∈G,∀a∈G,(a⋅e=a=e⋅a)

∀a∈G,∃b∈G,(a⋅b=e=b⋅a)

We can use a equivalent version of the definition, where the operation is associative and

∃e∈G,∀a∈G,(a⋅e=a)

∀a∈G,∃b∈G,(a⋅b=e)

But, I read in Herstein's book that if we consider the operation associative and with the properties

∃e∈G,∀a∈G,(a⋅e=a)

∀a∈G,∃b∈G,(b⋅a=e),

this is not equivalent to group definition