How Do You Determine the Torsion Coefficients of a Finite Abelian Group?

[Obraz35]

Homework Statement

Let G be a finite abelian group and let #(n) signify the number of elements x in G which satisfy x^n = e. Find the torsion coefficients of G when #(2)=16, #(4)=32, #(3)=9, #(9)=81 and x^36=e for all x in G.

Homework Equations

The Attempt at a Solution

I really am not sure how to approach this problem. We've been looking at how to classify finite abelian groups so I know that theorem should come into play, but how do I start figuring out what the coefficients are?...

Never mind, I figured it out...

 

[lippyka]

Let G be the finite abelian group. Let a,b,c,d be the torsion coefficients of G. Then, by the Fundamental Theorem of Finite Abelian Groups, G is isomorphic to a product of cyclic groups Z(a) x Z(b) x Z(c) x Z(d).Then, by Lagrange's Theorem, #(2)=16 = a^2, #(4)=32 = b^2 * c^2, #(3)=9 = a^3 * d^3, #(9)=81 = a^9 * b^9 * c^9 * d^9 and x^36=e for all x in G implies that a^3 * b^3 * c^3 * d^3 = 36. Solving this system of equations for a,b,c,d yields a=2, b=1, c=4, d=3. Thus, the torsion coefficients of G are a=2, b=1, c=4, d=3.