Question 1018437
For all *[tex \large x,y \in G], we have *[tex \large (x*y)*(x*y) = e] since x*y produces another element in the group.


If we consider the expression *[tex \large x*y*y*x], note that y*y = e, so the expression reduces to *[tex \large x*e*x = x*x = e]. So *[tex \large (x*y)*(y*x) = e] for all *[tex \large x,y \in G]


However since *[tex \large x*y] and *[tex \large y*x] are both inverses of x*y in G, we must have that *[tex \large x*y = y*x] for all x,y in G.