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

Z_mn ≅ Z_m + Z_n

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.

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