Florianópolis, Sept. 1999

10 × S3

Prove that all the groups below belong to the same isomorphism class.

  1. $ S_{3}$
  2. $ Aut(\mathbb{Z}_{2}×\mathbb{Z}_{2})$
  3. $ Aut(S_{3})$
  4. $ gen[ u(z)\!=\!e^{i2\pi/3}\!\cdot\!z \ ,\ v(z)\!=\!\bar{z} ]\quad\subset\quad\mathbb{C}^{\mathbb{C}}$.
  5. $ gen[ \begin{pmatrix}0&1\\ 1&0\end{pmatrix}\ \ ,\ \ \begin{pmatrix}1&0\\ 0&-1\end{pmatrix} ]\quad\subset\quad GL(2,\mathbb{Z})$
  6. $ D_{3}$
  7. $ gen[ f(x)=\frac{1}{1-x}\quad,\quad g(x)=\frac{1}{x} ]\quad\subset\quad\mathbb{R}[x]$
  8. $ G=\big(a,b\ ;\ a^{3},b^{2},(ab)^{2}\big)$
  9. $ GL(2,\mathbb{F}_{2})$
  10. The smallest possible non-Abelian group.

Notation $ gen[ U ]$ means group generated by the set U. Having the inclusion $ U\subset X$ in a group (or semi-group) X it should be clear what is the operation on the set.

$ A^A$ is the set of all possible functions $ f:A\longrightarrow A$.

$ \mathbb{K}[x]$ is the field of rational functions of the variable x over field $ \mathbb{K}$.

$ GL(n,\mathbb{K})$ is the group of all invertible matrices $ n×n$ with elements in field (ring) $ \mathbb{K}$.

$ (A\ ;\ B)$ denotes the presentation of the group: $ A$ is a set of (abstract) generators, elements of $ B$ together with their conjugates generate a subgroup that is annihilated (kernel of a homomorphism).