In my algebra class, we saw that every non-abelian group of order $6$ is isomorphic to $S_3$. I am curious as to whether this is true for higher orders. The natural answer seems that there are no such orders, but I don't know how to prove this. Thank you for your time.
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- Please formulate the question in the body and summarize it in the title 2. What do you mean with "isomorphic to one group G" ? Do you mean that there is exactly one non-abelian group of that order upto isomorphism ?
– Peter Oct 29 '23 at 08:11 -
2An infinite family of examples of such orders (probably not covering all cases) is $ord(G)=pq$ with primes $p<q$ such that $p\mid q-1$. – Peter Oct 29 '23 at 08:16
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1@Peter Indeed not all cases, see Martin Brandeburg's answer with $n=75$: there's no element of order $3$ in $GL(1,3)$ and only one conjugacy class of elements of order $3$ in $GL(2,3)$. – ancient mathematician Oct 29 '23 at 09:24
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I meant $GL(1,5)$ and $GL(2,5)$. – ancient mathematician Oct 29 '23 at 13:24
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@Peter I edited the question according to your suggestion. Thanks for the insights. – jayatra21 Oct 29 '23 at 14:37
1 Answers
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There are many more such numbers. They can be generated with the following piece of GAP code.
numberNonAbelianGroups := function(n)
local allGroups, abelianGroups;
allGroups := AllSmallGroups(n);
abelianGroups := AllSmallGroups(n,IsAbelian);
return Size(allGroups) - Size(abelianGroups);
end;
numbersWithOneNonAbelianGroup := function(limit)
local n;
return Filtered([1..limit],n -> numberNonAbelianGroups(n) = 1);
end;
The output of numbersWithOneNonAbelianGroup(100); is:
[ 6, 10, 14, 21, 22, 26, 34, 38, 39, 46, 55, 57, 58, 62, 74, 75, 82, 86, 93, 94 ]
This is the OEIS sequence A060650, about which probably not much is known. The related sequence A060689 counts the non-abelian groups of a given order.
As remarked in the comments, an infinite subsequence is formed by the products $p \cdot q$ of two primes with $p \mid q-1$ (see SE/881971). The examples $\leq 100$ are
[ 6, 10, 14, 21, 22, 26, 34, 38, 39, 46, 55, 57, 58, 62, 74, 82, 86, 93, 94 ]
so these only miss out $75$. The next examples in the sequence which are not products of two primes are $105, 165, 195, 231, 285, 357, 363, 385, 429, 483$ (all of which are products of three primes).
The sequence is contained in A060652, the orders for which at least one group is non-abelian. These can be classified explicitly by means of their prime factorization (see also SE/227628). So this is a necessary condition.
Martin Brandenburg
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