Groups of order 2023
How many groups are there with 2023 elements?
There's obviously at least one: Z2023, the integers mod 2023.
Now 2023 = 7 * 289 = 7 * 17 * 17 and so we could also look at
Z7 + Z17 + Z17
where + denotes direct sum. An element of this group has the form (a, b, c) and the sum
(a, b, c) + (a', b', c')
is defined by
((a + a)' mod 7, (b + b') mod 17, (c + c)' mod 17).
Is this a different group than Z2023? Are there any other groups of order 2023?
Let's first restrict our attention to Abelian groups. The classification theorem for finite Abelian groups tells us that there are two Abelian groups of order 2023:
Z7 + Z289
and
Z7 + Z17 + Z17
But what about Z2023? There's a theorem [1] that says
Zmn Zm + Zn
if and only if m and n are relatively prime. Since 7 and 289 are relatively prime, t
Z2023 Z7 + Z289.
The theorem also says that Z17 + Z17 is not isomorphic to Z289 and it follows that their direct sums with Z7 are not isomorphic.
So we've demonstrated two non-isomorphic Abelian groups of order 2023, and a classification theorem says these are the only Abelian groups. There are no non-Abelian groups of order 2023, though that's harder to show, and so we've found all the Abelian groups with 2023 elements.
More group theory posts[1] Sketch of proof. Let d be the greatest common divisor of m and n. If d > 1 then every element of Zm + Zn has order mn/d < mn and so Zm + Zn if cannot be isomorphic to Zmn. On the other hand, if d = 1, then Zm + Zn has an element of order mn and so is cyclic.
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