Abstract algebra cyclic subgroups

Click For Summary

Homework Help Overview

The problem involves a group G with exactly eight elements of order 10, and participants are discussing how many cyclic subgroups of order 10 exist within G.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are exploring the implications of having cyclic subgroups of order 10 and questioning how many elements of order 10 would necessitate the existence of such subgroups. There is also discussion about the sharing of elements between cyclic subgroups.

Discussion Status

Some participants have provided insights into the relationship between the number of elements of order 10 and the cyclic subgroups, while others are questioning assumptions about the sharing of elements among these subgroups. The discussion appears to be productive, with participants clarifying concepts and exploring different interpretations.

Contextual Notes

There is an underlying assumption regarding the properties of cyclic groups and the implications of the Euler phi function in relation to the number of elements of a given order.

tyrannosaurus
Messages
31
Reaction score
0

Homework Statement


Suppose that G is a group with exactly eight elements of order 10. How many cyclic subgroups of order 10 does G have?

Homework Equations





The Attempt at a Solution


I really don't have a clue how to solve this, any help would be greatly appreciated.
 
Physics news on Phys.org
If you have one cyclic subgroup of order 10, how many elements of order 10 does that force G to have?
 
you would need four elements of order 10 (by the Euler phi function). does that mean there would be two cyclicsubgroups of order2?
 
tyrannosaurus said:
you would need four elements of order 10 (by the Euler phi function). does that mean there would be two cyclicsubgroups of order2?

You tell me. You definitely need more than one. Can two cyclic subgroups of order 10 share any elements of order 10?
 
there would only be 2 cyclic subgroups of order 10 because non of the subgroups can share an order 10 element because if they did share an element in common, that element would generate both groups, so the two groups would be the same. So this means that no two cyclic subgroups of order 10 can share an element of order 10.
Thanks a lot for the help, this made sooooooo much more sense now.
 

Similar threads

Characterizing subgroups of a cyclic group
  • · Replies 9 ·
Replies
9
Views
2K
Group is a union of proper subgroups iff. it is non-cyclic
  • · Replies 1 ·
Replies
1
Views
5K
Cyclic group has 3 subgroups, what is the order of G
  • · Replies 5 ·
Replies
5
Views
3K
Every infinite cyclic group has non-trivial proper subgroups
  • · Replies 3 ·
Replies
3
Views
2K
Show that a group with no proper nontrivial subgroups is cyc
  • · Replies 1 ·
Replies
1
Views
3K
Proving that an Abelian group of order pq is isomorphic to Z_pq
  • · Replies 5 ·
Replies
5
Views
3K
Proving Normality of Subgroups in Cyclic Groups
  • · Replies 1 ·
Replies
1
Views
2K
Verifying a Proof about Maximal Subgroups of Cyclic Groups
  • · Replies 1 ·
Replies
1
Views
2K
Proving that Aut(K), where K is cyclic, is abelian
  • · Replies 5 ·
Replies
5
Views
2K
Groups that cannot be the direct product of subgroups
  • · Replies 1 ·
Replies
1
Views
2K