How Many Abelian Groups of Given Order

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In summary, the problem involves integrating the floor function with an exponential over the interval [0, infinity]. Various attempts at solving it using analytic techniques have been unsuccessful. One suggestion is to integrate over smaller intervals and then sum over those intervals, but the issue of not being able to integrate the function remains.
  • #1
darkchild
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Integrating Floor Function

Homework Statement


If [tex]\lfloor{x}\rfloor[/tex] denotes the greatest integer not exceeding x, then [tex]\int_{0}^{\infty}\lfloor{x}\rfloor e^{-x}dx=[/tex]

Homework Equations



none

The Attempt at a Solution



I don't know how to start this problem. At first, I tried bringing the floor function outside of the integral and using a summation because it only takes on discrete values, then I realized that those discrete values get multiplied by the exponential infinitely many times as x varies from any integer to the integer that is one greater.

I've tried the product rule. I looked up both the derivative and the indefinite integral of the floor function. If I use [tex]u=\lfloor{x}\rfloor[/tex], du=0 and I get back the original integral. If I use [tex]dv=\lfloor{x}\rfloor[/tex], the integral just gets more complicated.
 
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You aren't going to have much luck with the usual analytic techniques. Integrate it over the interval [n,n+1] for n an integer. So the floor function has a definite value. Then sum over n.
 
  • #3
Dick said:
You aren't going to have much luck with the usual analytic techniques. Integrate it over the interval [n,n+1] for n an integer. So the floor function has a definite value. Then sum over n.

The problem I'm having is that I can't integrate this at all. I'm not even at the step where I fiddle with the limits of integration/take sums yet.
 
  • #4
darkchild said:
The problem I'm having is that I can't integrate this at all. I'm not even at the step where I fiddle with the limits of integration/take sums yet.

You can't integrate a function like f(x)=n*e^(-x) where n is a constant? What's the integral for x=2 to x=3 of floor(x)*e^(-x)? Where I'm writing 'floor' for the function instead of the funny brackets.
 
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Related to How Many Abelian Groups of Given Order

1. How do you determine the number of Abelian groups of a given order?

The number of Abelian groups of a given order can be determined using the fundamental theorem of finite Abelian groups. This theorem states that every finite Abelian group is isomorphic to a direct product of cyclic groups. Therefore, to determine the number of Abelian groups of a given order, we need to find the number of ways the order can be expressed as a product of cyclic group orders.

2. Can the number of Abelian groups of a given order be infinite?

No, the number of Abelian groups of a given order is always finite. This is because the order of a group is always a positive integer, and the number of ways a positive integer can be expressed as a product of other positive integers is always finite.

3. Is there a formula for calculating the number of Abelian groups of a given order?

Yes, there is a formula for calculating the number of Abelian groups of a given order. This formula is known as the partition function and is denoted by p(n). It gives the number of ways a positive integer n can be written as a sum of positive integers, disregarding the order of the terms. However, for large values of n, this formula becomes very complex and is not practical to use.

4. Can two Abelian groups of the same order be isomorphic?

Yes, two Abelian groups of the same order can be isomorphic. This means that they have the same group structure, even though the elements may be labeled differently. For example, the groups of order 4, {0, 1, 2, 3} with addition modulo 4 and {0, 2, 1, 3} with addition modulo 4 are isomorphic.

5. Is there a relationship between the number of Abelian groups of a given order and the order itself?

Yes, there is a relationship between the number of Abelian groups of a given order and the order itself. This relationship is known as the Lagrange's theorem, which states that the order of a subgroup must divide the order of the group. Therefore, the number of Abelian groups of a given order must be a factor of the order itself.

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