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| Math Central | Quandaries & Queries |
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Question from Skye, a student: Suppose that * is an associative binary operation on a set S. Show that the set H={a E S such that a*x=x*a for all x E S} is closed under *. (We think of H as consisting of all elements of S that commute with every element in S.) |
Hi,
Suppose a and b are in S then to show S is closed you need to show that a*b is in S. By the definition of S you need to verify that for every x ε S, (a*b)*x = x*(a*b)
Here is how I would start
(a*b)*x
= a*(b*x) by the associative porperty of the operation *
= a*(x*b) since b ε S
Can you see how to continue?
If you need further help then please write back,
Harley
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