- #1

Zorba

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Partition the list of groups below into isomorphism classes

1.[tex]\mathbb{Z}_8[/tex]

2.[tex]\mathbb{Z}_8^*[/tex] (elements of Z_8 relatively prime to 8)

3.[tex]\mathbb{Z}_4 \times \mathbb{Z}_2[/tex]

4.[tex]\mathbb{Z}_{14} \times \mathbb{Z}_5[/tex]

5.[tex]\mathbb{Z}_{10} \times \mathbb{Z}_7[/tex]

6.[tex]D_{70}[/tex] Symmetries of a regular 35-gon.

7.[tex]\mathcal{P}\{1,2\}[/tex] with symmetric difference as operation.

8.Fourth roots of unity

9.[tex]\langle (12)(34),(13)(24)\rangle \le S_4[/tex]

10. The set generated by the matrix representations of the quaternions.

I don't feel confident about my answers, but I said that 2+9, 4+5, and the rest in separate classes.

I grouped them first by order and then mainly examined element orders, but I'm pretty sure I left some out. Any thoughts?