Ch2.#27. If a, b, c are elements of a group, solve axb = c for x and solve a−1 xa =
c for x.
Demonstration. This is primarily an exercise in working with the basic properties
of a group and, in particular, not making assumptions that aren’t true about the
group, eg. not assuming that elements commute.
If axb = c then we proceed as follows, noting that we want to isolate x by itself
on one side of the equation.
axb = c
a−1 axb = a−1 c
xb = a−1 c
xbb−1 = a−1 cb−1
(note we multiply both sides on the left bya−1 )
(note that since a−1 a = e then we me remove it from the left side)
(multiply both sides on the rightbyb−1 )
x = a−1 cb−1
For the other equation, we use the same idea.
a−1 xa = c
aa−1 xa = ac
xa = ac
xaa−1 = aca−1
x = aca−1
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