- #1
Happiness
- 679
- 30
If G is a finite set closed under an associative operation such that ax = ay forces x = y and ua = wa forces u = w, for every a, x, y, u, w ##\in## G, prove that G is a group.
What I attempted:
If we can prove that for every x ##\in## G, x##^{-1}## is also ##\in## G, then by the closure of the operation, the identity element is also ##\in## G, and we are done.
To show that x##^{-1}## is ##\in## G, we need to show that for any x, y ##\in## G,
ax##^{-1}## = ay##^{-1}## forces x##^{-1}## = y##^{-1}## and
x##^{-1}##a = y##^{-1}##a forces x##^{-1}## = y##^{-1}##
What I attempted:
If we can prove that for every x ##\in## G, x##^{-1}## is also ##\in## G, then by the closure of the operation, the identity element is also ##\in## G, and we are done.
To show that x##^{-1}## is ##\in## G, we need to show that for any x, y ##\in## G,
ax##^{-1}## = ay##^{-1}## forces x##^{-1}## = y##^{-1}## and
x##^{-1}##a = y##^{-1}##a forces x##^{-1}## = y##^{-1}##