Section 9.1 Algebraic Structures
697
Suppose that x, y, and z are members of a group 3G, # 4 and that x # z = y # z. To
conclude that x = y, we take advantage of z
−1
. Thus,
x # z = y # z
implies
(x # z) # z
−1
= ( y # z) # z
−1
x # (z # z
−1
) = y # (z # z
−1
)
x # i = y # i
x = y
Hence, G satisfies the right cancellation law.
PrACtiCe 18 Show that any group 3G, # 4 satisfies the left cancellation law.
PrACtiCe 19 Solve the equation x + 8 3 = 1 in 3Z 8 , + 8 4.
We have proved the following theorem.
tHeoRem on CAnCELLAtion in A GrouP
Any group 3G, # 4 satisfies the left and right cancellation laws.
tHeoRem on soLvinG LinEAr EquAtions in A GrouP
Let a and b be any members of a group 3G, # 4. Then the linear equations a # x = b
and x # a = b have unique solutions in G.
example 12
We know that 3Z 6 , # 6 4 is not a group. Here the equation
4 # 6 2 = 1 # 6 2
holds, but of course 4 ∙ 1.
Again, working in 3Z, + 4, we would solve the equation 6 + x = 13 by adding −6 to both sides, producing a unique answer of x = (−6) + 13 = 7. The
property of being able to solve linear equations for unique solutions holds in all
groups. Consider the equation a # x = b in the group 3G, # 4 where a and b belong
to G and x is to be found. Then x = a
−1 # b is an element of G satisfying the equation. Should x 1 and x 2 both be solutions to the equation ax = b, then a # x 1 = a # x 2
and, by left cancellation, x 1 = x 2 . Similarly, the unique solution to x # a = b is
x = b # a
−1
.
The theorem on solving linear equations tells us something about tables for
finite groups. As we look along row a of the group operation table, does element b
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