Today I Learned

Some of the things I've learned every day since Oct 10, 2016

104: Bijectivity of Left- or Right-Multiplication by Group Elements

Let G be a group and x be any element of that group. Then the functions

f: a \mapsto xa

g: a \mapsto ax

are bijective.

Proof:

Consider the function f. By the left-cancellation property of groups, if ab = ac, then b = c. This means that if b \neq c, then ab \neq ac, and it follows that f is injective. Since the domain and codomain of f are the same (G), this means that f is in fact bijective.

The proof for bijectivity of g is identical, but uses the right-cancellation property instead.

The intuitive significance of this is that left- or right-multiplication by a given element always sends distinct elements to distinct elements, which is pretty neat.

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