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Suppose $G$ is a finite group and I know for every $k \leq |G|$ that exactly $n_k$ elements in $G$ have order $k$. Do I know what the group is? Is there a counterexample where two groups $G$ and $H$ have the same number of elements for each order, but $G$ is not isomorphic to $H$? I suspect that there is, but I haven't thought of one.

Martin Sleziak
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Stanley
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  • Relevant: http://groupprops.subwiki.org/wiki/Order_statistics_of_a_finite_group –  May 24 '15 at 15:56
  • Oh, and that links to an answer: http://groupprops.subwiki.org/wiki/Finite_abelian_groups_with_the_same_order_statistics_are_isomorphic –  May 24 '15 at 15:56
  • (for abelian groups) –  May 24 '15 at 15:57
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    Two finite abelian groups with the same number of elements of each order are isomorphic, but in general the answer is no. – KCd May 24 '15 at 15:57
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    It links to an answer for finite groups in general http://groupprops.subwiki.org/wiki/Order_statistics-equivalent_finite_groups#Relation_with_group_properties. It appears that the answer is no. –  May 24 '15 at 15:58
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    See also http://math.stackexchange.com/questions/693163/groups-with-same-number-of-elements-of-each-order - this has been discussed before. – Nicky Hekster May 24 '15 at 16:39
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    It is generally believed that there is no list of properties of finite groups (such as orders of elements, orders of terms of central series, character tables, etc) which is sufficient to characterize groups up to isomorphism. – Derek Holt May 24 '15 at 17:38
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    @DerekHolt, look at http://arxiv.org/pdf/math/9210219.pdf. – KCd May 25 '15 at 00:23
  • See also http://math.stackexchange.com/questions/729611/can-we-uniquely-determine-a-group-given-the-orders-of-its-elements/ – Mikko Korhonen May 25 '15 at 16:26

3 Answers3

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Take $G=\mathbb{Z}/4\times \mathbb{Z}/4$, and $H=Q_8\times \mathbb{Z}/2$ of order $16$, where $Q_8$ denotes the quaternion group. Both groups have exactly $1$ element of order $1$, $3$ elements of order $2$ and $12$ elements of order $4$.

Edit: I understood the question as follows: Is there a counterexample where two groups $G$ and $H$ have the same number of elements for each order, but $G$ is not isomorphic to $H$ ? Is it really required, that all elements different from $1$ in $G$ have the same order ?

Hooked
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Dietrich Burde
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    You understood correctly. I wanted to know if the orders of the group elements fully characterise the group. I think you may have provided the smallest counterexample so far. – Stanley May 24 '15 at 18:07
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    The smallest counterexamples are of order 16, hence this is smallest possible (interestingly enough, there are 2 trios of 3 different groups that share same order statistics among the 14 groups of order 16, as well as another pair that share order statistics). – Justin Benfield Apr 30 '17 at 21:24
  • How to show $G$ and $H$ are not isomorphic? Thank you. – IncredibleSimon Feb 07 '25 at 13:25
  • @IncredibleSimon Hint: $G$ is abelian. – Dietrich Burde Feb 08 '25 at 12:57
37

Here is an example with two groups of order $27$. Consider the group $G$ which is elementary abelian (all elements $x \in G$ satisfy $x^{3} = 1$), of order $27$. And then $H$ the non-abelian group of order $27$ and exponent $3$ (once more, $x^{3} = 1$ for all $x \in H$). Concretely, $$ \left\{\, \begin{bmatrix} 1 & a & b\\ 0 & 1 & c\\ 0 & 0 & 1 \end{bmatrix} : a, b, c \in F \,\right\}, $$ where $F = \mathbb{Z}/ 3 \mathbb{Z}$ is the field with three elements.

Andreas Caranti
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Let $p$ be an odd prime. Let $G$ be the non-abelian group of matrices of the form $$\begin{pmatrix}1&a&b\\0&1&c\\0&0&1\end{pmatrix}\in\operatorname{GL}(3,\mathbb F_p).$$ Then $|G|=p^3$ and each element $g\ne 1$ has order $p$; this follows from the fact that $g-1$ is nilpotent, hence $(g-1)^p=(g-1)^3=0$ and finally $g^p=(1+(g-1))^p=1^p+(g-q)^p=1$.

Likewise the abelian group $H=(\mathbb Z/p\mathbb Z)^3$ is also of exponent $p$, i.e., all elements $\ne1$ have order $p$. As $H$ is abelian and $G$ is not, certainly $G\not\cong H$.

Nicky Hekster
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Hagen von Eitzen
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