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**Conn** Is there any simple way of telling if two groups are isomorphic? For example, given a list like this and asked to partition into isomorphism classes

1.$\displaystyle \mathbb{Z}_{4} \times \mathbb{Z}_{2}$

2.$\displaystyle \mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \mathbb{Z}_{2}$

3.$\displaystyle Q_{8}$

4.$\displaystyle D_{8}$

5.$\displaystyle \mathbb{Z}_{8}$

6.$\displaystyle \powerset \left\{ 1,2,3 \right\} $ under symmetric set difference $\displaystyle \left( X \cup Y \right) \backslash \left( X \cap Y \right) $

7. Solutions of $\displaystyle x^{8}-1=0 $ in $\displaystyle \mathbb{C}$

8.$\displaystyle \left< \left\( 24\right\), \left( 12 \right) \left( 34 \right) \right> \leq S_{4}$

9.$\displaystyle \left< \left[ {\begin{array}{cc} i & 0 \\ 0 & -i \\ \end{array} } \right], \left[ {\begin{array}{cc} 0 & i \\ i & 0 \\ \end{array} } \right], \left[ {\begin{array}{cc} 0 & -1 \\ 1 & 0 \\ \end{array} } \right] \right> \subseteq \mathbb{C}^{2\times2} $

10.$\displaystyle \mathbb{Z}_{16}^{*}$

Can I work out if they are isomorphic without listing out the elements of each, trying to find a map to from each one to another etc?