LS11-10 - S UMMARY OF 11/10 L ECTURE Given a group G and a...

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S UMMARY OF 11/10 L ECTURE Given a group G and a positive integer n , let ψ ( n ) = ± ± ± { g G : | g | = n } ± ± ± i.e. ψ ( n ) is the number of elements of G which have order n . Suppose that ψ ( n ) 1 for some n . Then there exists an element g of G such that | g | = n . We deduce that the elements 1 ,g,g 2 ,...,g n - 1 are distinct, and that g n = 1 . Note that each of these is a solution to the equation (1) x n = 1 . This is easy to check: for any k , we have ( g k ) n = g kn = ( g n ) k = 1 k = 1 . So, we’ve found n distinct solutions to equation (1). Intuitively, since the equation has degree n , we don’t expect there to be more than n solutions. For example, there are two distinct solutions to x 2 = 1 in the group Q × : x = ± 1 . The equation x 3 = 1 has only one solution in Q × , but three distinct solutions in the larger group C × . (Good review: what are these solutions? You should be able to write them down explicitly.) Unfortunately, our intuition is wrong. The Klein V group has four distinct solutions to the equation x 2 = 1 . However, our intuition exists for a reason – there are a number of groups which have the nice property that x n = 1 doesn’t have too many solutions (in particular, Q × , R × , C × all have this property). This motivates the following definition: Definition 1. A group G will be called nice if equation (1) has at most
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LS11-10 - S UMMARY OF 11/10 L ECTURE Given a group G and a...

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