When there is only one group of a given size

Today's date, US style, is 9/26/2023, and there is only one group, up to isomorphism, of size 9262023. You could verify this in Mathematica with the command

 FiniteGroupCount[9262023]

which returns 1.

For a given n, when is there only one group of size n?

There are two requirements. First, n has to be the product of distinct primes, i.e. no prime appears in the factorization with a power greater than 1. Second, no prime divides one less than another prime.

Now

9262023 = 3 * 41 * 257 *293

and you can check that 3 does not divide 40, 256, or 292, nor does 41 divide 2, 252, or 292, etc.

A more compact way to state the criteria above is to say

gcd(n, (n)) = 1

where (n) is Euler's totient function, the number of positive numbers less than n and relatively prime to n.

Why are these criteria equivalent? If

n = pqr...

then

(n) = (p - 1)(q - 1)(r - 1)...

Ifn and (n) have a nontrivial common factor, it has to be one of the prime factors of n, and none of these divide any term of (n).

Source: Dieter Jungnickel. On the Uniqueness of the Cyclic Group of Order n. The American Mathematical Monthly, Vol. 99, No. 6. (Jun. - Jul., 1992), pp. 545-547.

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