State the structure theorem for finite Abelian groups.
Up to isomorphism, what are the Abelian groups of order 360?
Solution
Write \(C_m=\mathbb Z/m\mathbb Z\) as an additive group.
1. Every finite abelian group has a decomposition\[ A\cong\bigoplus_{p\ \mathrm{prime}} \bigoplus_{j=1}^{s_p}C_{p^{e_{p,j}}}, \]where only finitely many \(s_p\) are nonzero and\(1\leq e_{p,1}\leq\cdots\leq e_{p,s_p}\).For each prime \(p\), the ordered list of exponents is uniquelydetermined by \(A\). Conversely, any such finite direct sum is afinite abelian group. Empty lists are allowed, including theempty sum for the trivial group. This is the elementary-divisorform of the structure theorem [@DF04].
2. Exactly six isomorphism classes have order \(360\).
| Group | Exponent |
|---|---|
| \(C_8\oplus C_9\oplus C_5\) | \(360\) |
| \(C_8\oplus C_3\oplus C_3\oplus C_5\) | \(120\) |
| \(C_4\oplus C_2\oplus C_9\oplus C_5\) | \(180\) |
| \(C_4\oplus C_2\oplus C_3\oplus C_3\oplus C_5\) | \(60\) |
| \(C_2\oplus C_2\oplus C_2\oplus C_9\oplus C_5\) | \(90\) |
| \(C_2\oplus C_2\oplus C_2\oplus C_3\oplus C_3\oplus C_5\) | \(30\) |
Factor \(360=2^3\cdot3^2\cdot5\). In step 1, the exponents for a prime \(p\) sum to the exponent of \(p\) in the group order. For \(p=2\) the sum must be \(3\). The only partitions of \(3\) into positive integers are \((3)\), \((1,2)\), and \((1,1,1)\), giving \[ C_8,\qquad C_2\oplus C_4,\qquad C_2\oplus C_2\oplus C_2. \] For \(p=3\) the only partitions of \(2\) are \((2)\) and \((1,1)\), giving \(C_9\) or \(C_3\oplus C_3\). The \(5\)-part is \(C_5\), and no other prime occurs. These three choices for the \(2\)-part and two choices for the \(3\)-part produce precisely the six rows. Each row has order \(8\cdot9\cdot5=360\), so every listed possibility is realized.
3. The six listed groups are pairwise nonisomorphic.
The exponent of a finite additive group is the least positive integer that annihilates every element, and is invariant under isomorphism. In a direct sum of cyclic groups, an integer kills every element exactly when it is divisible by the order of each cyclic summand. Thus the exponent is the least common multiple of those orders.
The least common multiples for the six rows are, respectively, \(360,120,180,60,90,30\). They are pairwise distinct, so no two rows are isomorphic. Combined with step 2, this proves both exhaustiveness and absence of repetitions.