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Cancellation laws

Left and right cancellation hold in every group

Proposition (Cancellation laws). Let GG be a and let a,b,c∈Ga,b,c\in G.

  • (Left cancellation) If ab=acab=ac, then b=cb=c.
  • (Right cancellation) If ba=caba=ca, then b=cb=c.
Equivalent characterizations

Equivalently, for each a∈Ga\in G, the translations La(x)=axL_a(x)=ax and Ra(x)=xaR_a(x)=xa are injective.

Remarks

Cancellation follows by multiplying by a−1a^{-1} on the appropriate side.