- #1
Mr Davis 97
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Homework Statement
Let ##A## and ##B## be finite groups, and ##A \times B## be their direct product. Given that ##(a,1)## and ##(1,b)## commute, and that ##(a,1)^n = (a^n,1)## and ##(1,b)^n = (1,b^n)## for all a and b, show that the order of ##(a,b)## is the least common multiple of the orders of a and b.
Homework Equations
The Attempt at a Solution
Let ##n## be the order of ##(a,b)##. Then ##(a,b)^n = (1,1)##,
##[(a,1)(1,b))]^n = (1,1)##
##(a,1)^n (1,b)^n = (1,1)##
##(a^n,1)(1,b^n) = (1,1)##
##(a^n,b^n) = (1,1)##
So ##a^n = 1##and ##b^n = 1##. It must be the case that n divides the order of ##a## and the order of ##a##. The smallest number that divides both is by definition the least common multiple.
Is this proof okay?