Partitioning into Isomorphism Classes: Exam Q&A

[Zorba]

Just had an exam there, one of the questions was

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?

 

[micromass]

Hi Zorba!

I hope your exam was a good one. Here are my thoughts on the matter:

Just had an exam there, one of the questions was

Partition the list of groups below into isomorphism classes

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

Cyclic group with 8 elements.

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

This group contains the elements {1,3,5,7}, thus the group has order 4. The only groups of order 4 are [itex]\mathbb{Z}_4[/itex] and [itex]\mathbb{Z}_2\times\mathbb{Z}_2[/itex]. Checking the orders of all elements gives us that all elements have order at most 2. Thus the group is [itex]\mathbb{Z}_2\times \mathbb{Z}_2[/itex].

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

Group of order 8. Not cyclic. Thus not isomorphic to 1.

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

Group of order 70. Abelian. Isomorphic to [itex]\mathbb{Z}_2\times\mathbb{Z}_5\times\mathbb{Z}_7[/itex].

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

Group of order 70. Abelian. Isomorphic to [itex]\mathbb{Z}_2\times\mathbb{Z}_5\times\mathbb{Z}_7[/itex]. Isomorphic to 4.

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

Group of order 70. Non-abelian. Not isomorphic to 4 or 5.

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

Group of order 4. Denote that A+A=0 for all elements A. Thus the group is isomorphic to [itex]\mathbb{Z}_2\times\mathbb{Z}_2[/tex]. Isomorphic to 2.

8.Fourth roots of unity

Group of order 8. Cyclic with generator [itex]i^{1/2}[/itex]. Thus is isomorphic to 1.

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

Group contains elements {(12)(34),(13)(24),(14)(23),1}. Group contains 4 elements and all elements have order at most 2. Isomorphic to [itex]\mathbb{Z}_2\times\mathbb{Z}_2[/itex]. Isomorphic to 2,7.

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

Group of order 8, non-abelian. Not isomorphic to 3,8

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?

I think the partition is: 2+7+9, 1+8, 4+5 and 3,6,10 in separate classes.

 

[Zorba]

Thanks for the reply, but I don't think 1 can't be iso to 8 because they have different orders. Also I toyed with including 7 with 2,9, but I didn't in the end, I think you're probably right though, for some reason I thought <{1}> has order 3, damn it...

 

[micromass]

Oh, I'm sorry, the fourth roots of unity are {1,-1,i,-i}. This is indeed cyclic of order 4. Thus not isomorphic to any other group...

 

[blue_raver22]

I would suggest reviewing the definitions and properties of isomorphism and isomorphism classes in group theory. In this case, isomorphism classes refer to groups that are isomorphic to each other, meaning they have the same structure and can be mapped onto each other in a one-to-one and onto manner. This means that the elements and operations in one group can be matched with the elements and operations in another group.

Looking at the list of groups given, it is important to note that the first three groups (1, 2, and 3) are all isomorphic to each other. This is because they are all cyclic groups of order 8, with elements of order 2. Similarly, groups 4 and 5 are both cyclic groups of order 70, and are therefore isomorphic to each other. Group 6, D_{70}, is a dihedral group of order 70, which is not isomorphic to any of the other groups listed.

Group 7, \mathcal{P}\{1,2\}, is the power set of the set \{1,2\} with symmetric difference as the operation. This group is isomorphic to the Klein four-group, which is the group \mathbb{Z}_2 \times \mathbb{Z}_2. Therefore, group 7 can be grouped with groups 1, 2, and 3.

Group 8, fourth roots of unity, is a subgroup of the multiplicative group \mathbb{C}^*. This group is isomorphic to \mathbb{Z}_4, and can be grouped with groups 1, 2, and 3.

Group 9, \langle (12)(34),(13)(24)\rangle \le S_4, is a subgroup of the symmetric group S_4. This group is isomorphic to the dihedral group D_4, and can be grouped with group 6.

Group 10, the set generated by the matrix representations of the quaternions, is the quaternion group of order 8. This group is isomorphic to groups 1, 2, and 3, and can be grouped with them.

In summary, the groups can be partitioned into 4 isomorphism classes:

1. \mathbb{Z}_8, \mathbb{Z}_8^*, \mathbb{Z}_4 \times \