There are two elements \(x,y\) in a group (\(G\),∗) such that every element in…
2014
There are two elements \(x,y\) in a group (\(G\),∗) such that every element in the group can be written as a product of some number of x's and \(y\)'s in some order. It is known that
\(x*x=y*y=x*y*x*y=y*x*y*x=e\)
where \(e\) is the identity element. The maximum number of elements in such a group is ____.
Attempted by 66 students.
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Correct answer: 4
Key idea: use the given relations to show x and y are involutions that commute, so the group generated by them has at most four elements.
From x*x = e and y*y = e we get x^{-1} = x and y^{-1} = y (x and y are involutions).
The relation x*y*x*y = e means (xy)^2 = e, so (xy)^{-1} = xy. But (xy)^{-1} = y^{-1}x^{-1} = yx, hence xy = yx. Thus x and y commute.
A group generated by two commuting involutions consists of {e, x, y, xy} because every product of x's and y's reduces to one of these four elements.
Some of these could coincide in special cases (for example if x = e or x = y), but the maximum possible number of distinct elements is 4.
Therefore the maximum number of elements in such a group is 4.