Let $G_1,G_2$ be two groups.We say that $G_1,G_2$ are similar iff for every integers $a_1,a_2,...,a_n\in \{1,-1\}$ and every function $f:\{1,...,n\}\rightarrow\{1,...,n\}$ we have the following:

$$\forall x_1,x_2,...,x_n\in G_1[x_{f(1)}^{a_1}x_{f(2)}^{a_2}...x_{f(n)}^{a_n}=e_1]$$ $$\iff \forall x_1,x_2,...,x_n\in G_2[x_{f(1)}^{a_1}x_{f(2)}^{a_2}...x_{f(n)}^{a_n}=e_2] $$

$e_1,e_2$ are the identities of $G_1,G_2$ respectively. Informally: The above definition says that for example if $G_1$ satisfies $x^2y^2zx^{-2}y^{-2}=e_1$ for every $x,y,z\in G_1$ and $G_2$ is similar to $G_1$, then $G_2$ satisfies $x^2y^2zx^{-2}y^{-2}=e_2$ for every $x,y,z\in G_2$ as well.

The additive groups $\mathbb{Z},\mathbb{Q}$ are two similar non-isomorphic groups.

Question: Are there two finite similar non-isomorphic groups ?

Robert Israel gave an answer for my above question. However, I would like to see an answer to the stronger question:

Are there two finite similar non-isomorphic groups of equal cardinality ?

In fact, this is the reason why I chose $\mathbb{Z},\mathbb{Q}$ and not $\mathbb{Q},\mathbb{R}$ in my counterexample of the infinite case.

  • $\begingroup$ If I'm reading your question right, doesn't this imply the groups would have the same presentations? $\endgroup$ – Alexander Gruber Apr 29 '14 at 15:27
  • $\begingroup$ @AlexanderGruber I don't know what the defintion of presentation is, I will check it now $\endgroup$ – Amr Apr 29 '14 at 15:29
  • $\begingroup$ "Presentation" is the wrong word. "Laws" is the right word. Different groups can have the same laws. $\endgroup$ – Jack Schmidt Apr 29 '14 at 15:35
  • $\begingroup$ A presentation gives equations that are valid for a certain set of generators, not necessarily for all members of the group. $\endgroup$ – Robert Israel Apr 29 '14 at 15:37
  • $\begingroup$ Agree. I just noted that presentation is not what I want $\endgroup$ – Amr Apr 29 '14 at 15:37

Consider ${\mathbb Z}_2 \times {\mathbb Z}_2$ and ${\mathbb Z}_2$. In both cases the only possible equations are where $ \{i: f(i)=j\}$ has even cardinality for all $j$.

  • $\begingroup$ Thanks. Is there an example where $G_1,G_2$ have the same cardinality? $\endgroup$ – Amr Apr 29 '14 at 15:37
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    $\begingroup$ One should be able to use Z2 x Z2 x Z4 and Z4 x Z4, I believe, though I don't have Neumann's book handy. I believe the variety generated by a finite abelian group of exponent e is the variety of abelian groups of exponent e. $\endgroup$ – Jack Schmidt Apr 29 '14 at 16:05
  • $\begingroup$ @JackSchmidt I think your sugesstion will work, I will try it. $\endgroup$ – Amr Apr 29 '14 at 16:19

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