Group implies quasigroup

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Statement

Verbal statement

Any group is a quasigroup.

Statement with symbols

Let G be a group, and a,b \in G be (not necessarily distinct) elements. Then, there exist unique x,y \in G satisfying ax = b and ya = b respectively.

Definitions used

Quasigroup

Further information: Quasigroup

A magma (S,*) (a set S with binary operation *) is termed a quasigroup if for any a,b \in S, there exist unique x,y \in S such that a * x = y * a = b.

Proof

Given: A group G, elements a,b \in G

To prove: There exist unique solutions to ax = b and ya = b

Proof: We have:

ax = b \implies a^{-1}(ax) = a^{-1}b  \implies x = a^{-1}b

Conversely:

x = a^{-1}b \implies ax = a(a^{-1}b) \implies ax = b

Thus:

ax = b \iff x = a^{-1}b

So, ax = b has a unique solution.

Similarly:

ya = b \implies (ya)a^{-1} = ba^{-1}  \implies y = ba^{-1}

Conversely:

y = ba^{-1}\implies ya = (ba^{-1})a  \implies ya = b

Thus:

ya = b \iff y = ba^{-1}

So, ya = b as a unique solution.