Partitioning into Isomorphism Classes: Exam Q&A

  • Context: Graduate 
  • Thread starter Thread starter Zorba
  • Start date Start date
  • Tags Tags
    Classes Isomorphism
Click For Summary
SUMMARY

The discussion centers on partitioning groups into isomorphism classes based on an exam question involving various mathematical groups. The groups analyzed include \(\mathbb{Z}_8\), \(\mathbb{Z}_8^*\), \(\mathbb{Z}_4 \times \mathbb{Z}_2\), and others. Key conclusions indicate that groups 2 and 9 are isomorphic, as are groups 4 and 5, while groups 1, 3, 6, 8, and 10 remain distinct. The final partition suggested is 2+7+9, 1+8, and 4+5, with the remaining groups in separate classes.

PREREQUISITES
  • Understanding of group theory concepts, including isomorphism and group order.
  • Familiarity with cyclic and abelian groups.
  • Knowledge of specific groups such as \(\mathbb{Z}_n\) and their properties.
  • Ability to analyze group structures and element orders.
NEXT STEPS
  • Study the properties of cyclic groups and their isomorphisms.
  • Learn about the classification of finite abelian groups.
  • Explore the concept of group actions and their applications in symmetry.
  • Investigate the structure of dihedral groups and their characteristics.
USEFUL FOR

Mathematics students, particularly those studying abstract algebra, group theorists, and educators looking to deepen their understanding of group isomorphism and classification.

Zorba
Messages
76
Reaction score
0
Just had an exam there, one of the questions was

Partition the list of groups below into isomorphism classes

1.\mathbb{Z}_8

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

3.\mathbb{Z}_4 \times \mathbb{Z}_2

4.\mathbb{Z}_{14} \times \mathbb{Z}_5

5.\mathbb{Z}_{10} \times \mathbb{Z}_7

6.D_{70} Symmetries of a regular 35-gon.

7.\mathcal{P}\{1,2\} with symmetric difference as operation.

8.Fourth roots of unity

9.\langle (12)(34),(13)(24)\rangle \le S_4

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?
 
Physics news on Phys.org
Hi Zorba! :smile:

I hope your exam was a good one. Here are my thoughts on the matter:
Zorba said:
Just had an exam there, one of the questions was

Partition the list of groups below into isomorphism classes

1.\mathbb{Z}_8

Cyclic group with 8 elements.

2.\mathbb{Z}_8^* (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 \mathbb{Z}_4 and \mathbb{Z}_2\times\mathbb{Z}_2. Checking the orders of all elements gives us that all elements have order at most 2. Thus the group is \mathbb{Z}_2\times \mathbb{Z}_2.

3.\mathbb{Z}_4 \times \mathbb{Z}_2

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

4.\mathbb{Z}_{14} \times \mathbb{Z}_5

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

5.\mathbb{Z}_{10} \times \mathbb{Z}_7

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

6.D_{70} Symmetries of a regular 35-gon.

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

7.\mathcal{P}\{1,2\} with symmetric difference as operation.

Group of order 4. Denote that A+A=0 for all elements A. Thus the group is isomorphic to \mathbb{Z}_2\times\mathbb{Z}_2[/tex]. Isomorphic to 2.<br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> 8.Fourth roots of unity </div> </div> </blockquote><br /> Group of order 8. Cyclic with generator i^{1/2}. Thus is isomorphic to 1.<br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> 9.\langle (12)(34),(13)(24)\rangle \leq S_4 </div> </div> </blockquote><br /> Group contains elements {(12)(34),(13)(24),(14)(23),1}. Group contains 4 elements and all elements have order at most 2. Isomorphic to \mathbb{Z}_2\times\mathbb{Z}_2. Isomorphic to 2,7.<br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> 10. The set generated by the matrix representations of the quaternions. </div> </div> </blockquote><br /> Group of order 8, non-abelian. Not isomorphic to 3,8<br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> I don&#039;t feel confident about my answers, but I said that 2+9, 4+5, and the rest in separate classes.<br /> I grouped them first by order and then mainly examined element orders, but I&#039;m pretty sure I left some out. Any thoughts? </div> </div> </blockquote><br /> I think the partition is: 2+7+9, 1+8, 4+5 and 3,6,10 in separate classes.
 
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...
 
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...
 

Similar threads

MHB Subgroups of the dihedral group D6
  • · Replies 9 ·
Replies
9
Views
12K
MHB Is my solution for this exercise correct?
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad Dfns group action & group, written in terms of the other
  • · Replies 26 ·
Replies
26
Views
1K
MHB Abelian Groups of Order $2100$: Elements of Order $210$
  • · Replies 1 ·
Replies
1
Views
2K
Theorem of Finitely Generated Abelian Groups
  • · Replies 3 ·
Replies
3
Views
2K
Abelian group as a direct product of cyclic groups
  • · Replies 9 ·
Replies
9
Views
2K
Undergrad Base Pi Integers: Isomorphism with Rationals?
  • · Replies 33 ·
2
Replies
33
Views
4K
MHB Finding Isomorphisms between Groups: D6, A4, S3xZ2, G
  • · Replies 1 ·
Replies
1
Views
2K
MHB Matrix rings acting on right R-modules - Dauns - Exercises 1-5 no 2
  • · Replies 4 ·
Replies
4
Views
2K
Challenge Math Challenge - June 2023
  • · Replies 80 ·
3
Replies
80
Views
10K