Math Is Fun Forum

Stanley_Marsh

Member

Registered: 2006-12-13

Posts: 345

Question

Is it true that

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Numbers are the essence of the Universe

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Question

No. The set of all nonzero real numbers under multiplication can’t be written this way.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Question

What exactly do you mean multiplicative?  Typically, any group operation is written multiplicatively.

If that's the case, neither can finite non-cyclic groups, such as the group of symmetries of the square (D8 (or D4, depends on your notation) ) or the group of permutations of order 6 (S6).

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Stanley_Marsh

Member

Registered: 2006-12-13

Posts: 345

Re: Question

Here is the question
Show that if G is a finite group of even order , then G has an odd number of elements of order 2. Note that e is the only element of order 1.

If I regard G as 

The proof will be very easy , but I am not sure whether I can do it this way.

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Numbers are the essence of the Universe

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Question

No, you can’t do it that way.

Here is one way to do it. Notice that the elements of order 2 are their own inverse. e is also an element that is its own inverse. The other elements are not their own inverse. If an element is not its own inverse, that element and its inverse must form a distinct pair – hence the number of such elements must be even (so that they can be neatly paired up with each other). So we have

|G| = (number of elements that are their own inverse) + (number of elements that are not their own inverse)
      = 1 + (number of elements of order 2) + (number of elements that are not their own inverse)

Since |G| is even, the number of elements of order 2 must be odd for the equation above to hold.

Last edited by JaneFairfax (2007-05-16 04:34:32)